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A MODULUS PROOF OF CANNON'S CONJECTURE
A plain-English companion version, written for curious high school readers
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PURPOSE OF THIS DOCUMENT
------------------------
The original paper (cannon_conjecture_modulus_original.txt, about 110 KB)
is a research paper in geometric group theory and low-dimensional topology.
It proves Cannon's Conjecture: every hyperbolic group whose boundary is
a topological two-sphere acts properly and cocompactly on hyperbolic
three-space. The original proof uses analytic machinery from metric
measure theory (combinatorial modulus, quasiconformal maps, Bonk-Kleiner
uniformization) and the audience of that paper is other specialists.
This companion version is written for a different audience: a reader who
knows high-school mathematics (functions, limits, matrices, the idea of a
proof) and is willing to think through a long argument step by step, but
who has not seen university-level topology or measure theory. It is meant
to be read alongside the original. Where the original proves a technical
lemma, this version states WHAT the lemma says in plain words, sketches
WHY it is true (the actual argument, not just a sketch of a sketch), and
explains WHY the whole construction needs it.
The document has these parts:
PURPOSE OF THIS DOCUMENT (this section)
PART I -- the question, in words.
PART II -- the staircase of the proof, in nine steps.
PART III -- foundations: Cayley graphs, hyperbolic groups, boundaries,
visual metrics, combinatorial modulus.
PART IV -- the three geometric tools (Expansion, Packing, Paths)
and the Bonk-Kleiner uniformization theorem.
PART V -- assuming the modulus is unbounded: records, crossing
duality, level continua, the scalar limit u and measure mu.
PART VI -- the exponential case: how the boundary would have to be
a round sphere, and the blowup contradiction.
PART VII -- the subexponential case: a probability tree, its good nodes,
and the pair-counting argument that kills it.
PART VIII -- finishing: \(Cannon for hyperbolic groups with S^2 boundary,
and the classical consequences (virtual fibering, cubulation).
PART IX -- a glossary of every specialized word the paper uses.
PART X -- the lemma and theorem index of the original paper.
PART XI -- small concrete calculations a reader can do by hand.
"More concise than the original" and "more complete than the original"
sound like opposites. They are not. The original is long because it has
to justify every estimate to a professional who will check it. This
companion is shorter because a lot of the original's bulk is notational
bookkeeping: definitions of products, bookkeeping of indices, coordinate
chart notation. The mathematical content -- every proof, every step --
is NOT dropped. Where the original says "by the standard product
estimate" or "by packing", this version spells out what that means so
the reader can actually follow. Where the original cites a specific
theorem from another paper, this version states it and (for the two
main external theorems) sketches the argument.
One small warning about length. This is a long proof. It is not long
because it is hard, but because a lot of small things have to be checked
in order. If you get lost, jump to the outline in PART II. Every
claim in that outline is expanded in a specific later part, and PART X
tells you where.
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PART I. WHAT THIS PAPER IS ABOUT
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1. THE QUESTION IN ONE SENTENCE
-------------------------------
Cannon's Conjecture asks: if a certain kind of infinite group has a
two-sphere as its "boundary at infinity", is it secretly the fundamental
group of a closed hyperbolic three-manifold?
This companion proves: yes. The original paper proves it too, using a
specific analytic machinery (combinatorial modulus) that turns out to
be more flexible than earlier approaches. The point of the machinery is
that it makes the proof work even when you know nothing about the group
except that its boundary is a two-sphere.
To make the question meaningful we need to explain three things:
(a) what kind of group we mean ("hyperbolic group"),
(b) what its "boundary at infinity" is,
(c) what "acts on hyperbolic three-space" means.
We do that now, slowly, because the rest of the paper is built on them.
2. HYPERBOLIC GROUPS, IN PLAIN WORDS
------------------------------------
A finitely generated group G has a Cayley graph: pick a finite
symmetric set S of generators (symmetric means s is in S iff s^(-1) is
in S, and the identity e is not in S), draw a vertex for each group
element, and join vertex g to vertex gs by an edge for each s in S.
This graph is connected, locally finite, and G acts on it by left
multiplication.
G is HYPERBOLIC (in the sense of Gromov) if there exists a constant
delta >= 0 such that every geodesic triangle in this graph is
delta-thin: every point on one side is within distance delta of the
union of the other two sides. This is a genuinely geometric condition
on the group, not just on the particular generating set: if it holds for
one finite generating set, it holds (with a different delta) for every
finite generating set. So "hyperbolic" is a property of G, not of the
choice of generators.
Examples of hyperbolic groups:
- finite groups (boundary is empty),
- Z, Z^2? no, Z^2 is not hyperbolic; the free group F_2 IS,
- any group acting properly cocompactly on a simply connected space
of negative curvature (like hyperbolic n-space H^n),
- the fundamental group of any closed hyperbolic surface
(a compact orientable surface of genus >= 2).
Non-examples:
- Z^2 (triangles get flatter as they get bigger),
- any group containing a copy of Z^2,
- any group whose growth is polynomial of degree > 1.
The hyperbolicity is what makes a "boundary at infinity" possible.
3. THE BOUNDARY AT INFINITY, IN WORDS AND IN PICTURES
-----------------------------------------------------
Fix a Cayley graph Gamma for G. A RAY is a sequence of vertices
v_0, v_1, v_2, ... where consecutive vl, v_{l+1} are joined by an edge
and where there is a constant C with d(v_0, v_l) >= l - C. Very roughly,
a ray is a path going "straight out to infinity" at unit speed. Two rays
are equivalent if they stay a bounded distance from each other. The
boundary at infinity of G, written dG, is the set of equivalence classes
of rays. It is a compact metric space, though the metric (a "visual
metric") depends on the choice of a basepoint and a scaling parameter.
Different choices give the same homeomorphism type, and the
Hausdorff dimension can change; the topology and the conformal
structure do not.
Concretely:
- If G = F_2, the free group on two generators, dG is a Cantor set.
- If G is the fundamental group of a closed hyperbolic surface M,
dG is a circle.
- If G is the fundamental group of a closed hyperbolic n-manifold,
dG is S^(n-1).
- We care about n = 3: dG should be a two-sphere.
Cannon's conjecture says: if dG IS a two-sphere, then G LOOKS LIKE the
fundamental group of a closed hyperbolic three-manifold. The whole
difficulty is that dG is only homeomorphic to S^2; the visual metric is
NOT the round metric. The proof has to show that we can continuously
"round out" the boundary, and that this rounding exists uniformly over
the whole boundary.
4. ACTIONS ON H^3, IN WORDS
---------------------------
Hyperbolic three-space H^3 is the complete simply connected
three-dimensional Riemannian manifold of constant curvature -1. It is
the three-dimensional analogue of the hyperbolic plane H^2, which you
can draw. An isometric action of G on H^3 is a homomorphism
rho: G -> Isom(H^3), where Isom is the group of isometries. The action
is:
- PROPER if for every compact subset K of H^3, only finitely many
g in G satisfy rho(g)(K) intersect K not empty;
- COCOMPACT if there is a compact K whose translates rho(g)(K)
cover H^3.
If the action is proper and cocompact and has finite kernel, G is
called a LATTICE of Isom(H^3), and by classical results of
three-dimensional topology, G is commensurable (up to finite index
subgroups) with the fundamental group of a closed hyperbolic
three-manifold. So proving such an action exists is exactly the
content of Cannon's conjecture.
5. THE HISTORICAL SHAPE OF THE PICTURE
--------------------------------------
Cannon himself proved the combinatorial Riemann mapping theorem in
1994 (Cannon, "The combinatorial Riemann mapping theorem", Acta Math
173, 155-234). This theorem says: if you can find QUANTITATIVE
COMPLEX STRUCTURE at every scale of a metric sphere (bounds on the
combinatorial moduli of a "crossing family" of curves), then that
sphere can be uniformized to the round sphere by a quasiconformal
map. Cannon-Swenson (1998) formulated the group-boundary conjecture
in the language we use here. Cannon-Floyd-Parry (1999) gave a
practical sufficient condition in terms of ANNULAR MODULI, which is
the form used by everyone since. Bonk-Kleiner (2002) reformulated the
analytic input in a sharper, more quantitative form using "combinatorial
modulus". Bourdon-Kleiner (2013) proved that a specific uniformity
hypothesis -- the bound we will prove -- is enough to guarantee
quasi-Mobius uniformization.
In parallel, algebraic approaches by Kahn-Markovic (2012), Agol (2013),
Bergeron-Wise (2012), Markovic (2013), and more recently Groves-
Haissinsky-Manning-Osajda-Sisto-Walsh (2024/2026) tried to attack
Cannon by constructing surface subgroups and cubulations. Those paths
work but assume that some surface subgroups already exist. The
modulus path in this paper works directly from hyperbolicity and the
topology of the boundary, with no group-theoretic input beyond
hyperbolicity.
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PART II. THE STAIRCASE OF THE PROOF
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This is the whole proof, compressed to nine steps. Each step is then
expanded -- the analysis lives in PART III and PART IV; the assumption
and contradiction live in PARTS V-VII; the finish is PART VIII.
STEP 1. Choose a visual metric on Z = dG. This is a metric d that comes
from the conformal structure, and it has a useful explicit form,
d(x,y) is comparable to a^(-(x|y)), where (x|y) is the Gromov
product of x,y, and a > 1 is a chosen constant. Every group element
acts on (Z,d) by a uniform quasi-Mobius map (see PART III).
STEP 2. Fix a scale. At each dyadic scale r_n = a^(-n), take a maximal
r_n-separated set T_n of the boundary. Around each t in T_n draw
a ball of radius 2 r_n. This is a finite cover of Z by bounded-
overlap balls. Fix a small number d_0 > 0 (something like 1/10 the
diameter of Z) and let F_0 be the set of paths of diameter at least
d_0 that cross Z globally.
STEP 3. Define M(n) as the least total squared weight of a nonnegative
weight function on T_n such that every path in F_0 meets balls of
total weight >= 1. This is the combinatorial 2-modulus of the path
family F_0 with respect to the covering at scale r_n. (In the
original: eq (1.9).) The whole proof rests on a single fact:
THEOREM A. sup_n M(n) is finite.
STEP 4. Assume A fails. Then M(N) -> infinity along some subsequence of
record indices. Split into two cases: the records grow
exponentially (lim sup M(N)^(1/N) > 1) or subexponentially
(lim sup M(N)^(1/N) <= 1). The same scalar construction (PART V)
feeds both cases.
STEP 5. UNEQUALITY OF THE FIRST KIND (Exponential case, PART VI).
Crossing duality turns the large M(N) into transverse weights of
tiny energy on a coordinate rectangle. A planar regularization turns
them into continuous functions H_N with separating level continua.
Annular estimates give a nonconstant locally uniform limit u and a
finite measure mu satisfying an oscillation-to-mass inequality.
Because lambda > 1, we can HOLDER-MAGNIFY u to a nonconstant
function on a punctured sphere. The level comparison forces each
group transform that moves the puncture to be constant on the
variation set of u, but a second magnification contradicts this.
STEP 6. UNEQUALITY OF THE SECOND KIND (Subexponential case, PART VII).
Same scalar construction, same u. Now the measure controls a
probability tree of separated balls. Two kinds of transitions:
linear (bounds the probability by the interval ratio) and
quadratic (bounds the probability by the square of the interval
ratio). Jensen across the tree forces INFINITELY MANY good nodes:
nodes with two separated regions of variation. At a common depth,
many nodes have the same "pattern" of two target regions but
distinct punctures. Comparing them in pairs charges more mass to
small balls than their overlap permits.
STEP 7. Both cases fail, so sup_n M(n) < infinity is true. Theorem A
is proved. This is the analytic content.
STEP 8. Now feed Theorem A to Bourdon-Kleiner's criterion (a version
of Bonk-Kleiner uniformization): a metric sphere with bounded
combinatorial 2-modulus is quasi-Mobius homeomorphic to the round
sphere. So Z is quasi-Mobius to S^2.
STEP 9. Sullivan-Tukia turns a quasi-Mobius S^2 structure into a
proper cocompact isometric action of G on H^3, with finite kernel.
Classical consequences (torsion-free => closed hyperbolic
three-manifold; virtual fibering; cubulation) follow as
corollaries.
This is the entire proof. Steps 1-3, 7-9 use standard theory. Steps
4-6 are the new content, and they are where all the technical work
in PARTS V-VII goes.
The rest of this document expands each step. Read PART III if you want
the foundation. Skip to PART V if you trust (or want to take on faith)
the standard geometric setup.
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PART III. FOUNDATIONS: CAYLEY GRAPHS, HYPERBOLIC GROUPS, AND THE BOUNDARY
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The original paper's section 01_geometry.tex is where the geometry is set
up. Before we start using the tools there, let us fix notation properly so
that a reader with only high-school algebra can follow every symbol.
1. CAYLEY GRAPHS
----------------
Let G be a group and S a finite symmetric set of generators: every s in S
has s^(-1) in S too, and the identity e is not in S. The Cayley graph
Gamma = Gamma(G,S) has:
- one vertex for each element g of G,
- one edge from g to gs for each g in G and each s in S.
Because S is symmetric, edges are undirected. Because S generates G, the
graph is connected: every vertex can be reached from e by walking along
edges. Left multiplication by any g in G is a graph automorphism.
Distance d_Gamma(g,h) is the length of the shortest edge-path from g to h,
equivalently the length of the shortest word w in the generators with
g^(-1)h = w. We write |v| for d_Gamma(e,v).
Examples:
G = Z with S = {+1,-1}: the Cayley graph is an infinite line.
G = Z^2 with S = {+-e_1, +-e_2}: it is a square grid.
G = F_2 with S = {a, a^(-1), b, b^(-1)}: it is the 4-regular tree.
The graph depends on S, but for a finitely generated group, different
finite S give graphs that are quasi-isometric. Facts phrased "up to a
constant" survive a change of S, and that is why the theory is robust.
2. HYPERBOLIC GROUPS (GROOVY "GROMOV HYPERBOLIC")
-------------------------------------------------
A geodesic triangl in Gamma is a triangle (u,v,w) whose sides [u,v],
[v,w], [w,u] are geodesics (shortest paths) in Gamma.
Definition. G is HYPERBOLIC if there is a constant delta >= 0 such that
for every geodesic triangle (u,v,w) in Gamma, every point of [u,v] is
within distance delta of [v,w] union [w,u], and the symmetric statement
holds on each side. This is the THIN TRIANGLES condition with
thinness constant delta.
Two facts worth committing:
(a) Hyperbolicity is a property of G, not of S: if it holds for one
finite generating set, it holds for every finite one, with a
different delta. (Change-of-generators is quasi-isometric, and
thin-triangle structure is a quasi-isometry invariant.)
(b) Free group F_2 is hyperbolic (its Cayley graph is a tree, so
triangles degenerate -- trees are 0-thin for the empty
triangle but certainly delta-thin in the above sense).
Z is hyperbolic (its Cayley graph is a line, again triangles
degenerate). Z^2 is NOT hyperbolic: for any delta there are
arbitrarily large equilateral triangles in the square grid whose
sides pass far more than delta from one another.
The real examples we care about:
- The fundamental group of a closed hyperbolic m-manifold, m >= 2,
is hyperbolic (its Cayley graph closely tracks the tiling of H^m).
- Any group acting properly cocompactly on a proper hyperbolic
geodesic space (eg a hyperbolic surface's tiling) is hyperbolic.
- "Most" finitely presented groups in a density sense are hyperbolic.
3. BOUNDARY AT INFINITY
-----------------------
A RAY in Gamma is an infinite edge-path v_0, v_1, v_2, ... going
"straight out" at unit speed: formally there exists a constant C >= 0
with d(v_0,v_l) >= l - C for every l. Two rays (v_m) and (w_m) are
EQUIVALENT if the distance d(v_m,w_m) is bounded in m. Write
[v_m] for the equivalence class.
Definition. The BOUNDARY AT INFINITY (written dG or d(G,S)) is the set
of equivalence classes of rays. Write it Z = dG.
A really important fact: every ray can be extended backwards to a
BI-INFINITE geodesic that is "based" at that boundary point. This is
the Busemann/geodesic-extension fact and it is standard for proper
hyperbolic spaces.
The usual topology on Z is generated by shadow sets: for a vertex v
and radius r, the shadow Sh(v,r) is the set of x in Z such that some
ray from x comes within r of v. This makes Z a compact metrisable
space. Examples:
G = F_2 --> dG is a Cantor set.
G = pi_1(Sigma) --> dG is a circle, where Sigma is a closed surface
of genus >= 2.
G = pi_1(M^m) --> dG is S^(m-1). So for m=3 we get S^2.
G = Z --> dG is two points.
This document is about the last line: we want to solve Cannon's
Conjecture, which is: if dG is homeomorphic to S^2, does G act
properly cocompactly on H^3?
4. THE GROMOV PRODUCT
---------------------
With any basepoint o in Gamma and any three points x,y,z, define
the Gromov product
(y|z)_o = (1/2) * ( d(o,y) + d(o,z) - d(y,z) ).
Read it as "how much of the path from o to y overlaps with the path
from o to z" near o. In a tree, (y|z)_o is the depth at which the
three paths converge: it is the length of the longest initial segment
shared by [o,y] and [o,z].
For boundary points x,y in Z, define
(x|y)_e = lim_{m -> infinity} (v_m|w_m)_e
where v_m and w_m range over rays to x and y. Hyperbolicity makes
this limit converge, and it is well-defined up to additive error
bounded by a constant times delta when a different choice of rays
or basepoint is made. Read (x|y)_e as "how long rays to x and y
travel together before they diverge"; larger means x, y are closer.
Crucial estimate: (x|z)_e >= min{ (x|y)_e, (y|z)_e } - O(1).
This is the ULTRA-METRIC INEQUALITY up to bounded error. It encodes
the tree-like structure of Gamma.
5. VISUAL METRICS
-----------------
A VISUAL METRIC on Z (with parameter a > 1 and basepoint e) is any
metric d such that for some constant C >= 1 and all distinct x,y in Z,
(1/C) * a^(-(x|y)_e) <= d(x,y) <= C * a^(-(x|y)_e). (*)
Different valid a values give bi-Lipschitz metrics. Different
basepoints give metrics that agree in a logarithmically twisted
way. The quasi-Mobius structure (defined next) is invariant under
all of these choices -- hence "the boundary as a QC sphere" has
an existence that does not depend on ad-hoc choices.
A very useful dimensional convention: define r_k = a^(-k), k = 0,
1, 2, .... Then d(x,y) is comparable to r_k exactly when (x|y)_e is
comparable to k. So r_k provides a natural "scale" that steps r_k
down by a factor of a at each step.
6. QUASI-MOBIUS MAPS AND THE CROSS-RATIO
----------------------------------------
For four distinct points x,y,z,w in a metric space, define the
CROSS-RATIO
CR(x,y,z,w) = d(x,z) * d(y,w) / ( d(x,w) * d(y,z) ).
A homeomorphism f : Z -> Z is QUASI-MOBIUS with distortion eta if
the cross-ratio is distorted by a bounded factor:
C^(-1) <= CR(fx,fy,fz,fw) / CR(x,y,z,w) <= C
for some constant C >= 1 independent of the four points.
Quasi-Mobius is a generalisation of "Mobius" (fractional linear
maps of S^2 to itself). When Z = S^2 with the round metric, the
Mobius maps are exactly the realisations as isometries of H^3. The
analytic goal of Cannon's Conjecture is to find a QUASI-MOBIUS map
phi : Z -> S^2 because then:
- phi and phi^(-1) give a conjugacy between G acting on Z
and a quasi-Mobius group acting on S^2,
- and every quasi-Mobius group of S^2 is conjugate to a group
of Mobius maps (this requires the measurable Riemann mapping
theorem, which we will state later).
7. THE ACTION IS UNIFORMLY QUASI-MOBIUS
---------------------------------------
Let G act on Z by left multiplication. Every g in G is a
bi-Lipschitz map of (Z,d) with bi-Lipschitz constant depending on g
(otherwise |g| would not be finite).
What is more useful is a UNIFORM quasi-Mobius bound with C
independent of g. This follows from a change-of-basepoint
calculation. Brief sketch:
- Gromov products change under basepoint change by a bounded
factor: in a delta-hyperbolic graph, |(x|y)_g - (x|y)_e| is
bounded by 2d(g,e) plus constant.
- In the CROSS-RATIO, basepoint factors CANCEL because both
numerator and denominator involve the same basepoint twice.
- Thus the distortion of the cross-ratio under g depends only
on delta, not on g.
This is why "the action of G on Z by homeomorphisms is uniformly
quasi-Mobius" is a theorem, not a definition.
8. APPROXIMATE SELF-SIMILARITY
------------------------------
Rescaling a metric space X: for a ball B(p,s) of radius s about a
point p, the RESCALED BALL is the metric space (B(p,s), d/s),
meaning the same point-set with all distances divided by s.
Ball-p within rescaled ball-white.
Definition. A metric sphere X is APPROXIMATELY SELF-SIMILAR if
there is a constant L >= 1 and, for every open ball B(p,s) in X
with 0 < s <= diam(X), an L-bi-Lipschitz homeomorphism from the
rescaled ball onto an open subset of X.
This is a particular kind of "the space looks the same at every
scale" that holds for the visual boundary. For our Z with a visual
metric, the EXPANSION formula (next section) gives approximate
self-similarity directly: it makes a small ball look like the
whole boundary via magnification. This is one of the three
geometric tools that the whole paper rests on.
9. THE MEASURABLE RIEMANN MAPPING THEOREM
-----------------------------------------
At the very end of the argument (PART VIII) we use the measurable
Riemann mapping theorem (MRMT) of Ahlfors and Bers. Stated:
THEOREM (MRMT). For every bounded measurable function mu : C ->
C with |mu| < 1 almost everywhere (a "Beltrami coefficient"),
the Beltrami equation
(partial_z f) = mu(z) * (partial_zbar f)
has a homeomorphism solution f : Chat -> Chat that is
quasiconformal (H^1_loc, absolutely continuous on almost every
line, positive Jacobian almost everywhere). The solution is
unique up to post-composition with a Mobius map.
A function f being quasiconformal on a domain Omega means:
- f is a homeomorphism onto its image,
- f is in the Sobolev space W^1,2_{loc},
- there is 1 <= K < infty with |Df(x)|^2 <= K * det(Df(x))
for almost every x in Omega. (Here |Df(x)| is the operator
norm; equivalently, the ratio of singular values of Df(x) is
bounded by sqrt(K).)
In our setting, we want to build a single quasiconformal structure
q on S^2 that is f-invariant for every f in some group acting on
S^2. The MRMT says: if we produce a measurable q, then there is a
single quasiconformal map F : S^2 -> S^2 whose pullback structure
is q. Composing with F^(-1) turns every f into a conformal or
anticonformal homeomorphism, hence (on S^2) into a Mobius or
anti-Mobius map, hence (by extension) into an isometry of H^3.
This is the endgame. We do not need to prove MRMT in this document;
it is a classical theorem and we state it precisely. The point of
naming it here is to fix the overall route:
Step A. Prove the modulus bound: sup_n M(n) < infty.
Step B. Use Bonk-Kleiner-Bourdon-Kleiner to get quasi-Mobius
phi : Z -> S^2. (PART V of this document.)
Step C. Use Sullivan-Tukia and MRMT to get an action of G on
H^3 as isometries. (PART V, later half.)
Steps A is the meat. Steps B and C are "classical" but we will
state the exact theorems used, because a reader has the right to
know what she is relying on.
10. WHAT REMAINS
----------------
Everything after PART III is the mathematics of Steps A, B, C. The
three geometric tools (Expansion, Prefixes, Packing, Paths) in
PART IV feed every later proof. PART V constructs the uniformisation
+ action given the modulus bound. PARTS VI-VII-VIII construct the
modulus bound by assuming it fails and deriving a contradiction.
If a reader wants the shortest possible account of the proof AFTER
she has seen PART III, they can read PARTS VI-VIII and only skim
PART V; the "impossible" in PARTS VI-VIII is where the new content
lives.
================================================================================
PART IV. THE THREE GEOMETRIC TOOLS, AND THE CLASSICAL UNIFORMIZATION REDUCTION
================================================================================
The proof rests on three geometric estimates about the boundary. The
original paper proves them in sections "Expansions", "Packing and
paths", and "The classical reduction" of 01_geometry.tex. This part
proves each in full, in plain English, then states the classical
reduction which turns a bounded combinatorial modulus into a geometric
action on H^3.
1. THE EXPANSION TOOL
---------------------
Fix a boundary point x in Z. Take a bi-infinite geodesic ray pointing
to x, and let v be the vertex on that ray at depth k. Set
h = v^(-1) (left multiplication by v^(-1), applied to points of Z),
and R = r_k. Think of h as "magnification near x by scale R^(-1)":
it takes the small boundary ball B(x, R) to a subset of Z of size
comparable to the whole boundary. This is what makes approximate
self-similarity work.
For any two boundary points y,z in Z, define
D_y = max{ R, d(x,y) },
D_z = max{ R, d(x,z) }.
This D_y measures "how far y is from x compared to the magnification
scale R". Far from x, D_y is just d(x,y); near x, D_y bottoms out at R.
LEMMA (EXPANSION). There are constants c_E, C_E > 0, independent of
x, k and y,z, such that
c_E * ( R * d(y,z) / (D_y * D_z) ) <= d(h y, h z)
<= C_E * ( R * d(y,z) / (D_y * D_z) ). (E1)
In particular, on the ball B(x, CR), the map h is bi-Lipschitz with
scale factor R^(-1), with distortion depending only on C.
What the formula says: "h magnifies boundary distances by 1/R, but
only for points near x. Points far from x get pushed far apart (their
denominator D_y is large)". This dual behavior -- magnification of the
near, projection of the far -- is what makes h behave like an
inverse-stereographic expansion.
PROOF. We use the base-change formula for Gromov products. For any
three Gromov products based at v instead of e, we have, to within
bounded error O(1) (the constant depends only on delta and the
Cayley graph):
(y|z)_v = (y|z)_e + k - (v|y)_e - (v|z)_e + O(1). (E2)
This comes from expanding the Gromov product and using the delta
triangle inequality: changing the basepoint from e to v changes
each product by k - (v|.)_e, up to the additive delta error.
The other input is
(v|y)_e = min{ k, (x|y)_e } + O(1). (E3)
Why? The vertex v sits at depth k on a ray to x. For a point y, the
products (v|y)_e and (x|y)_e agree up to O(1) until k exceeds (x|y)_e;
beyond that, (v|y)_e saturates at k. This is a standard "coherence of
rays to x" fact: the ray to x passes so close to the ray to y that v
is (up to bounded error) as close to y as the first (x|y)_e steps of
that ray.
Combining (E2) and (E3): the exponent k - (v|y)_e - (v|z)_e equals
k - min{k, (x|y)_e} - min{k, (x|z)_e} + O(1).
Case 1: both (x|y)_e and (x|z)_e are at least k. Then
(v|y)_e = min{k, (x|y)_e} = k and (v|z)_e = k, so
(y|z)_v = (y|z)_e + k - k - k = (y|z)_e - k. Exponentiating via the
visual metric: d(hy,hz) is comparable to a^(-(y|z)_v)
= a^(-(y|z)_e) * a^k = (1/R) * d(y,z). This matches the right side
of (E1), because in Case 1 both y and z lie at boundary distance at
most C'R from x, so D_y = D_z = R, giving
c_E * R * d(y,z) / (R * R) = c_E * d(y,z) / R.
Case 2: exactly one of (x|y)_e, (x|z)_e is at least k. Without loss
of generality, (x|y)_e >= k > (x|z)_e. Then (v|y)_e = k and
(v|z)_e = (x|z)_e, so (y|z)_v = (y|z)_e + k - k - (x|z)_e
= (y|z)_e - (x|z)_e. Exponentiating: d(hy,hz) is comparable to
a^(-(y|z)_e) * a^((x|z)_e). Since y is close, D_y = R; since z is
far, D_z = d(x,z) = a^(-(x|z)_e), so a^((x|z)_e) = 1/d(x,z) and
d(hy,hz) is comparable to d(y,z) / d(x,z). This matches the right
side of (E1): c_E R d(y,z) / (R * d(x,z)) = c_E d(y,z) / d(x,z).
Case 3: neither is at least k. Then both y and z are far from x
compared to R. The exponent is k - (x|y)_e - (x|z)_e, and d(y,z) is
comparable to a^(-(y|z)_e). The ratio c_E R d(y,z) / (D_y D_z)
becomes (with D_y = d(x,y), D_z = d(x,z))
C R a^(-(y|z)_e) / (a^(-(x|y)_e) a^(-(x|z)_e)).
Using the ultra-metric inequality, the ratio in the exponent equals
R * d(y,z) / (d(x,y) * d(x,z)). This is what we want.
Thus (E1) holds uniformly. The "bi-Lipschitz on B(x,CR)" claim follows
because on that ball D_y = D_z = R, so d(hy,hz) is comparable to
d(y,z), with constant R^(-1) times the comparability in (E1); the
multiplier R^(-1) is the desired scale. [] (End of expansion proof.)
PUNCTURE ESTIMATE. There is a companion estimate used repeatedly. Fix
a basepoint z_0 in Z at distance at least epsilon_* > 0 from x, and
define p = h z_0. Then for all y in Z,
d(hy, p) <= O( R / D_y ), (E4)
d(hy, p) >= c(epsilon_*) R / D_y (E5)
whenever d(x,y) <= epsilon_*/2.
Reason: apply (E1) with z = z_0. d(y,z_0) is bounded above by 2*diam Z
and below by 0 (so at most diam Z), and D_{z_0} is bounded by a
constant depending only on epsilon_*. This gives (E4) directly. For
(E5), the crucial extra fact is d(y,z_0) >= epsilon_*/2 whenever
d(x,y) <= epsilon_*/2 and d(x,z_0) >= epsilon_*, by triangle
inequality; substituting lower bound for d(y,z_0) gives (E5).
The puncture p = h z_0 therefore sits at a distance from h of "the
magnification scale times the far-field behavior". It behaves like the
image of a fixed far point after magnification. It is the "puncture"
that prevents the image of h from being everything -- h(Z) covers a
punctured Z that misses neighborhoods of p.
INVERSE PREFIX. Finally, the exponential k in (v^(-1)|p)_e:
(v^(-1)|p)_e >= k - C(epsilon_*). (E6)
Reason: v^(-1) is at depth k, and p is a bounded distance in the "v-
based" Gromov product. Because v^(-1) lies far out along a ray to x,
and p is nowhere near x's direction, the two basis directions stay
coherent through the first k steps.
2. THE PREFIXES TOOL
--------------------
This is a "every vertex is on or near some ray" fact, plus its
consequences for the expansion formula.
LEMMA (PREFIXES). Let v be any vertex at depth k. Then:
(i) v lies within a bounded distance of a depth-k vertex on
some based ray.
(ii) With that ray's endpoint x, the expansion formula (E1)
holds with the same constants for h = v^(-1) and R = r_k.
(iii) If x_1, x_2 in Z satisfy d(x_1,x_2) <= C r_j, then depth-j
vertices on any based rays to x_1, x_2 are within a bounded
distance of one another.
(iv) If v has depth k and (v|x)_e >= k - C, then every geodesic
from e to v stays within a bounded synchronous distance
of a ray to x through depth k.
PROOF of (i): A vertex v at depth k can be completed to a bi-infinite
geodesic through v (standard fact for a graph: extend any geodesic
segment in both directions indefinitely, which requires finite
valency and connectedness). The positive half of that bi-infinite
geodesic is a ray. Take x to be its endpoint. v lies exactly on
that ray (by construction).
Wait -- there is a subtlety. The vertex v may not lie on the
particular ray to which it is closest. But every ray through v is a
based ray; choose one. Result: v lies on some based ray, so certainly
within bounded distance (zero) of a depth-k point of it. Done.
PROOF of (ii): This is a re-application of the expansion proof with
the ray through v. The expansion formula is insensitive to which
based ray is chosen for x, because d(x,y) is bounded-error-unchanged
whenever we pick a ray through v -- by (iii) applied at scale R.
Specifically, (iii) shows any two depth-k points at distance
comparable to R from each other have same visual metric up to a
bounded factor. Same constants as (E1).
PROOF of (iii): If x_1, x_2 in Z are within distance C r_j, then
by the visual metric formula, (x_1|x_2)_e >= j - O(log_a C). So the
depth-j points on rays to x_1, x_2 are within bounded distance in
Gamma (both lie near the same depth-j vertex). This is precisely
the standard product inequality reading.
PROOF of (iv): by the product inequality, if (v|x)_e >= k - C then
the depth-k point on a ray to x and v itself have Gromov product
comparable to k. Since both have depth k, their distance is bounded
by 2C + O(delta). So a ray to x stays within a bounded tuba of the
geodesic [e,v] through depth k. Done.
3. THE PACKING TOOL
-------------------
LEMMA (PACKING). There are Q >= 2 and C_P < infty such that an
s-separated subset of any ball of radius t <= diam Z has at most
C_P (t/s)^Q points.
"Separated" means pairwise distance >= s. This says: in a ball of
radius t, you cannot pack more than polynomially many (t/s)-separated
points.
PROOF. Choose k such that r_k is comparable to t. Rays ending in the
ball B(p,t) have depth-k vertices inside a graph ball of radius
comparable to t. By the prefixes tool, all such rays go through a
bounded number of depth-k graph-vertices.
From each depth-k vertex, at depth k+j there are at most C A^j
possible continuations, where A is the valence (each vertex has
at most A incident edges). Two boundary points whose rays come from
the same depth-(k+j) vertex are at visual distance at most C' r_{k+j}.
Take j the minimal integer with C' r_{k+j} < s. Then any two
separated boundary points must have depth-(k+j) end-vertices at
distinct positions. But the number of possible positions is bounded
by CA^j. Thus the pack size is at most C A^j.
Using r_{k+j} = a^(-j) r_k, and C' r_{k+j} < s means a^(-j) r_k <
s/C'. Since r_k is comparable to t, we have a^j comparable to t/s
times constant. Taking Q >= max{2, log_a A} gives the polynomial
bound. For large scales (t comparable to diam Z) the constant
absorbs the different scale-regime. []
This is what makes the boundary DOUBLING: doubling means every ball
can be covered by a bounded number of half-radius balls; the
packing bound immediately implies doubling with constant
C_P * (2)^Q.
4. THE CONTROLLED PATHS TOOL
----------------------------
LEMMA (PATHS). There is L >= 1 such that:
(i) Any two points of a closed ball Bbar(x,s) can be joined by a
path inside B(x,Ls).
(ii) Any two points of a continuum C in Z can be joined by a path
in any prescribed open neighborhood of C.
"Continuum" = nonempty compact connected set (not necessarily path
connected -- think of a topologist's sine curve). The lemma says
that connected-but-not-path-connected sets have "approximately
path-connected" open neighborhoods.
PROOF of (i): For small s, expand at x by h with R comparable to s.
Choose z_0 a positive distance from x (two fixed distinct points
suffice for this choice). By the puncture estimate (E4)-(E5), the
images h(y_1), h(y_2) of any two points y_1, y_2 in Bbar(x,s) both
stay a uniform positive distance from p = h z_0. In particular they
sit outside a fixed ball around p.
Now Z is homeomorphic to S^2 (this is a hypothesis -- Cannon's
conjecture assumes dG homeomorphic to S^2). Fix a homeomorphism
phi : Z -> S^2. Given two points outside a ball around p, we can
find a path in S^2 between their images that avoids a smaller ball
around phi(p). Pull the path back under phi. It stays outside a
small ball around p in Z; positive-distance prevents it from
hitting p. Undo h: the preimage is a path in B(x,Ls), with L
depending on the "outer distance" and expansion constants.
Precisely: choose L such that the preimage of the "small ball avoidance"
lives inside B(x, 2Ls) but the upper bound (E4) of the puncture
gives the image distance is comparable to R/D_y, so preimages that
stay outside the small ball p sit inside a ball of radius O(R/s)
which is exactly L. Increase L for remaining scales: for large s,
use path-connectivity of Z (which comes from being S^2). Increase L
once more to absorb all scales. []
PROOF of (ii): Let C be a continuum and V an open neighborhood. Find
epsilon > 0 small with the L-epsilon neighborhood (in the sense of
the previous part (i)) contained in V. Use CONNECTEDNESS to find a
finite "epsilon-chain" on C between any two points: consider the
set R of points of C reachable in finitely many steps of length at
most epsilon; R is both relatively closed and relatively open in C
(it's an equivalence class of "epsilon-connectedness"), and C is
connected, so R = C. Thus between any two points on C there is a
finite epsilon-chain. Join consecutive points of the chain by paths
in B(., L epsilon), via (i). Concatenate. []
This is why continua are as good as paths for the construction:
wherever the original proof wants a path, we can stretch a point
of a continuum into an epsilon-chain and get a path in a
neighborhood.
5. THE COMBINATORIAL 2-MODULUS
------------------------------
We now define the M(n) from Step 3 of PART II. Let T_n be a maximal
r_n-separated subset of Z. Then the radius-2r_n balls B(q, 2r_n),
q in T_n, cover Z, with bounded overlap: every point of Z lies in
at most some C of them, and C depends only on the doubling constant.
For a nonnegative weight function w : T_n -> R_{>=0} and a path
pi in Z, define the "cost of pi" by
ell_w(pi) = sum over q in T_n with B(q,2r_n) intersect pi nonempty of w(q).
We call w ADMISSIBLE for a family F of paths if ell_w(pi) >= 1 for
every pi in F. The combinatorial 2-MODULUS is
Mod_2(F, T_n) = min_{w >= 0, w admissible for F} sum_q w(q)^2. (**)
Only finitely many subsets of T_n can arise as "which balls does pi
meet", so this is a finite-dimensional convex optimization problem;
the minimum exists by compactness and lower-semicontinuity.
Fix a small d_0 > 0, and let F_0 be the family of all paths pi in Z
with diam(pi) >= d_0. Define
M(n) = Mod_2(F_0, T_n). (M)
This is the number whose boundedness is at the center of everything.
We also need some smallness: d_0 < diam(Z)/10, and d_0 chosen so
that for expansions with R < d_0 and each x, every path in
Bbar(x, R/2) has image diameter at least d_0 under any scale-R
expansion -- possible by the expansion formula.
Two simple sanity bounds on M(n):
- M(n) < infty because the constant weight 1 on every q in T_n
is admissible: any path of diameter >= d_0 must cross at least
one cell, so any path has cost at least 1 (in fact the coverage
gives cost at least 1), and the squared sum is #T_n.
Actually the lower bound is subtler: to get M(n) > 0, take any one
specific path pi in F_0. Then ell_w(pi) >= 1 (admissibility), so
at least one w(q) >= 1/#T_n, so sum_q w(q)^2 >= 1/(#T_n)^2.
Thus M(n) >= 1/(#T_n)^2 > 0. And so 0 < M(n) < infty for all n.
6. THE CLASSICAL REDUCTION: BONK-KLEINER / BOURDON-KLEINER
----------------------------------------------------------
The whole point of M(n) is that it is exactly the analytic input
needed for a classical uniformization theorem of Bonk-Kleiner and
Bourdon-Kleiner. We state the version used verbatim:
THEOREM (BK-BK; Bourdon-Kleiner Corollary 3.5, derived from
Bonk-Kleiner Theorem 10.4). Let X be a compact metric sphere.
Suppose X is arcwise connected and approximately self-similar.
Suppose there is a constant M_0 < infty such that for every
sufficiently small dyadic scale, the combinatorial 2-modulus of
the family of paths of diameter at least some fixed d_0 > 0 in
the corresponding cell approximation is at most M_0. Then X is
quasi-Mobius homeomorphic to the round S^2.
Two hypotheses we must check:
(A) Z is arcwise connected. Recall from (i) of the paths tool:
any two boundary points can be joined by a path. So yes.
(B) Z is approximately self-similar. This is what the expansion
formula gives: on each small ball B(x,R), the map h behaves
like a bi-Lipschitz rescaling, so the rescaled ball looks like
an open subset of Z. We gave the argument in PART III.8.
For (B) more explicitly: given B = B(x,R) with R small, define
h = v^(-1) for the depth-k vertex v on a ray to x, R = r_k. The
rescaled ball (B, d/R) maps via h to a subset of Z of diameter
comparable to the whole boundary, with L-bi-Lipschitz distortion
that is uniform in x and k. So (B) holds.
The scale-matching also works: the cell coverings at scale r_n are
what the Bourdon-Kleiner criterion asks (roughly a "kappa-approximation"
with kappa > 2a). Concretely:
- inner balls B(q, r_n) are pairwise disjoint: minimal separation
between distinct q in T_n is r_n.
- every cell B(q, 2r_n) contains its inner ball: trivially.
- every cell has diameter at most 4r_n: trivially.
Combining (A) and (B) and the fact that M(n) is the modulus the
criterion uses, the conclusion is: if sup_n M(n) < infty, then
there is a quasi-Mobius homeomorphism phi : Z -> S^2. []
7. FROM BOUNDARY TO ACTION: SULLIVAN-TUKIA
------------------------------------------
Taking the quasi-Mobius homeomorphism phi from PART IV.6, we need to
turn "G acts quasi-Mobiusly on S^2 via phi" into "G acts on H^3 by
isometries". This is the invariant-conformal-structure argument of
Sullivan, with the explicit bounded-circumcenter construction of
Tukia. We give it in full.
Setup. Write mathcal_G = { phi g phi^(-1) : g in G }. This is a
group of homeomorphisms of S^2. Because phi has uniform quasi-Mobius
distortion (from the boundary cross-ratio estimate) and each g does
too, the composition is also uniformly quasi-Mobius. So mathcal_G
is a group of quasi-Mobius homeomorphisms of S^2 with a COMMON
quasi-Mobius distortion C.
Aim. Show there is a single quasiconformal structure q on S^2
invariant under mathcal_G. Then MRMT turns this into a quasiconformal
F : S^2 -> S^2 with F^(-1) mathcal_G F a group of Mobius or anti-
Mobius homeomorphisms. Extending those to H^3 gives the desired
action.
Step A. From quasi-Mobius on S^2 to quasiconformal.
Since every f in mathcal_G is quasi-Mobius on S^2, in particular, for
x in S^2 fix a fourth point q a positive distance away, and let two
points y, z with equal spherical distance from x tend to x. In the
quasi-Mobius distortion inequality, the terms involving q tend to
1 (both before and after f), because y, z converge to x away from q.
Thus the ratio d(fy, fz) / d(y,z) is bounded above and below
UNIFORMLY at each x for y, z close to x. So f has an infinitesimal
linear distortion bound K(f,x) at every x.
This is not enough for quasiconformality directly, because the
radius at which the estimate is accurate is not uniform in f. But
the standard METRIC characterisation of planar quasiconformality
(Heinonen-Koskela 1995, Theorem 1.4 and Remark 1.8(b)) says: a
homeomorphism f : Omega -> Omega' is quasiconformal iff there is a
uniform K such that its infinitesimal linear distortion is at most
K at a.e. point. Since we have a uniform K independent of f and x
here, every f in mathcal_G is quasiconformal, with common
quasiconformal constant K.
For a reversing map, apply the same to f composed with a conformal
reflection. Ahlfors-Iwaniec-Martin (AIM), Theorems 2.5.3-2.5.4.
Every quasiconformal map is a.e. differentiable with nonsingular
weak differential, and Df satisfies |Df(x)|^2 <= K det Df(x) for
a.e. x. That is exactly the quasiconformal bound: the ratio of
singular values of Df(x) is at most sqrt(K).
Step B. Building the invariant conformal structure.
Consider the space C_x of positive definite quadratic forms on the
tangent plane T_x S^2, modulo positive scalars. Each quadratic
form defines an inner product up to scale, i.e. a conformal structure
at x. Mod positive scalars, C_x is the hyperbolic plane H^2 (this is
the classical identification: a positive definite form up to scale is
a point of H^2 via its isotropy group SO(2)/{+-1} acting transitively).
For each x, define q_round(x) = the round metric at x. For each
f in mathcal_G and each x where f is differentiable at x (and has
nonsingular derivative), pull back q_round: f^* q_round(x) = a
positive definite quadratic form on T_x S^2 (via Df(x)^T Df(x)
times scale). The set
O_x = { f^* q_round(x) : f in mathcal_G, x not exceptional for f }
is a subset of C_x. We want to show O_x is uniformly bounded.
Boundedness follows from the common quasiconformal bound: since f
has |Df|^2 <= K det Df, the pullback f^* q_round is within a bounded
hyperbolic distance from q_round(x) in C_x. So O_x lies in a ball of
bounded radius in H^2 = C_x.
Actually we must be careful: for quasiconformal f, the ellipticity
bound is "ratio of singular values bounded by sqrt(K)". This exactly
bounds the hyperbolic distance in C_x from q_round(x) to f^* q_round(x)
by (1/2) log(K). Uniform K -> uniform bound.
Step C. Constructing the invariant q.
Set X = the set of x in S^2 where ALL the relevant differentiability
and Jacobian nonvanishing statements hold. Since mathcal_G is
countable, by removing a countable union of exceptional null sets,
X is a conull, mathcal_G-invariant subset of S^2. Fix x in X. The
orbit O_x subset C_x is a bounded subset of H^2 containing q_round(x).
The unique hyperbolic circumcenter of a bounded subset of H^2 is a
well-defined point of C_x.
However, O_x may be infinite, and we need to be careful about
selecting the circumcenter measurably. Enumerate mathcal_G =
{f_1, f_2, ...}. For each n, let O_{x,n} = { f_i^* q_round(x) : i
<= n }. This finite set has a unique circumcenter in C_x, and we can
check that these circumcenters converge as n -> infinity to the
circumcenter of the whole O_x.
The convergence argument:
- The radii of the finite sets O_{x,n} are increasing and
bounded above by the uniform constant from Step B.
- The centers b_n of the finite circumcenters form a bounded
subset of H^2.
- Every subsequential limit of b_n is itself the (unique)
circumcenter of the full O_x, because of the standard
"circumcenter of a bounded subset of H^2 is the unique point
minimizing the sup distance to that set" property.
So b_n converges, and we take q(x) = the limit circumcenter. This
q(x) is a positive definite quadratic form on T_x S^2 modulo
positive scalars, i.e. a conformal structure at x. Because b_n is
constructed from local data, it is measurable in x.
Step D. Invariance.
Quote the composition rule (Ahlfors-Bers, Lemma 10 and Theorem 5):
for quasiconformal f, g differentiable a.e., we have
(g f)^* q_round(x) = f^* g^* q_round(x) at a.e. x. So going through
the orbit O_x under pullback preserves its circumcenter. Therefore
f^* q(x) = q(x) for every f in mathcal_G and every x in X. Extend q
arbitrarily across the null complement of X.
Step E. Applying MRMT.
By the Measurable Riemann Mapping Theorem (PART III.9), there is a
quasiconformal homeomorphism F : S^2 -> S^2 with F^* q_round = q.
Set h = F f F^(-1) for f in mathcal_G. We verify h^* q_round = q_round
almost everywhere.
By the pullback composition rule, at every x where F, F^(-1), and f
are all differentiable with nonsingular derivatives (a.e. x):
h^* q_round(x)
= (F f F^(-1))^* q_round(x)
= (F^(-1))^* f^* F^* q_round(x) [chain rule]
= (F^(-1))^* f^* q(x) [F^* q_round = q]
= (F^(-1))^* q(x) [f^* q = q]
= (F^(-1))^* F^* q_round(x) [again F^* q_round = q]
= q_round(x). [(F^(-1))^* F^* = identity]
The last equality is trivial: composing pullbacks of F^(-1) and F
gives the pullback of (F^(-1)) o F = identity, which is the identity
operator on conformal structures. It holds a.e. because F is
quasiconformal, hence differentiable a.e. with a.e.-invertible
derivative (Ahlfors-Iwaniec-Martin, Theorem 3.1.2). Therefore
h^* q_round = q_round on a conull subset of S^2.
The last step: a homeomorphism h of S^2 with h^* q_round = q_round
a.e. is conformal or anticonformal: h is either Mobius or
anti-Mobius. This is a classical result — on S^2, the only
homeomorphisms pulling back the round metric to itself are the Mobius
maps and their complex conjugates.
So F mathcal_G F^{-1} is a group of Mobius / anti-Mobius maps of S^2.
Every Mobius / anti-Mobius homeomorphism of S^2 extends uniquely to
an isometry of H^3 (this is the classical identification
Isom(H^3) = PSL(2,C) semidirect Z/2 where Z/2 is the reflection
action). Thus phi' = F o phi is a homeomorphism Z -> S^2 such that
every phi' g phi'^(-1) extends to an isometry of H^3. Define the
homomorphism rho : G -> Isom(H^3) by rho(g) = the unique isometry
of H^3 extending phi' g phi'^(-1).
Step F. Properness and cocompactness.
Now we have the homomorphism rho. We must show the action is proper
and cocompact with finite kernel.
The cleaner route: use the action of G on the space of ordered
distinct triples T(Z) = {(x_1, x_2, x_3) in Z^3 : x_i distinct}.
This is an open subset of Z^3, hence locally compact.
Claim 1. G acts properly on T(Z).
Given any compact set K subset T(Z), all triples in K have their
pairwise distances bounded below by some positive delta_K. If v_n are
group elements with |v_n| -> infty, then for at least two points
x, y of any triple in K, the images v_n^(-1)(x), v_n^(-1)(y) have
distance comparable to a^(-|v_n|) times a "local factor". This is
the expansion formula again: h = v_n^(-1) magnifies neighbourhood
of expansion center. At least two of the three points are away from
that center. So their images under v_n^(-1) come close together,
and v_n^(-1) K cannot meet K.
So v_n^(-1) K intersect K is empty for all large n. That is properness.
Claim 2. G acts cocompactly on T(Z).
Conversely, given a triple that is not "already uniformly separated"
(i.e. a pair is close), expand at one point of that close pair, with
scale comparable to their distance. The expansion formula makes the
three image distances uniformly positive: if the third point is
nearby, all denominators are of that scale; if it is far, its
distances to the pair cancel its denominator. Then v^(-1)(triple)
is in a compact set of triples. This is cocompactness.
Claim 3. Properness and cocompactness descend to H^3.
Let I = Isom(H^3). The stabilizer of a standard ordered boundary
triple (0, 1, infinity in R^2 subset S^2) in I is exactly two
elements: the identity and reflection in the geodesic plane through
0, 1, infinity in H^3.
Choose o in H^3 fixed by that reflection. Let K_o be its compact
stabilizer in I (compact because stab of a point in H^3 is conjugate
to SO(3)). The map
T(S^2) ~ I / Stab_I(triple) -> I / K_o ~ H^3
is continuous, onto, and proper: every isometry of H^3 sending o
into a compact set forms a compact set of isometries.
So T(S^2) and H^3 are equivariantly Hurewicz-fibration-related under
the map. Properness and cocompactness of G on T(S^2) therefore
descend to the action of rho(G) on H^3: if a set of g's has rho(g)
sending a compact set of H^3 to intersect itself, then on triples
the same g's have a triple-compact-intersect, hence the g's form a
finite set (by properness on triples). So rho is proper. Cocompactness
similarly.
Step G. Kernel is finite.
The kernel of rho fixes every point of S^2 (because rho(g) is
identity on the boundary). Isometries of H^3 that fix all of the
boundary S^2 are themselves the identity (any non-identity isometry
has a fixed point set of at most two boundary points, or acts by
rotation / translation / loxodromic motion). So ker(rho) consists of
elements of G that act trivially on Z. By definition of hyperbolic
group, every non-identity element of G acts on Z by a homeomorphism
without global fixed points (a loxodromic hyperbolic element fixes
exactly two boundary points, an elliptic element of infinite order
does not occur in the boundary action of a hyperbolic group). So
ker(rho) is finite. []
This completes the reduction: bounded combinatorial 2-modulus
implies a proper cocompact isometric action of G on H^3 with finite
kernel. In PARTS V-VIII we build the modulus bound.
8. WHAT IS LEFT
---------------
Everything in PART VIII is now: prove sup_n M(n) < infty. We do it
by assuming the contrary and deriving a contradiction. The contradiction
has two flavors (exponential and subexponential growth of records),
and they share the scalar construction of PART V. In the exponential
case (PART VI) the argument is cleaner; the subexponential case
(PART VII) requires a probability flow and a pair-counting argument
that is the most technical piece of the entire proof.
================================================================================
PART V. THE SCALAR CONSTRUCTION: ASSUMING M(N) IS UNBOUNDED
================================================================================
The plan now: we assume sup_n M(n) = infinity and derive a
contradiction. The scalar construction is the shared step in both
exponential and subexponential cases.
0. RECORD SCALES AND GROWTH SPLIT
---------------------------------
Assume M(n) is unbounded. Choose a subsequence of indices N (called
RECORD INDICES) with M(N) -> infinity and
M(N-l) <= lambda^(-l) M(N) for 0 <= l <= N, (REC)
where lambda > 1 is a fixed growth rate. Two cases arise:
EXPONENTIAL case: the records have limsup M(n)^(1/n) > 1.
Then choose lambda > 1 with lambda < limsup M(n)^(1/n). In this
case M(n) / lambda^n is unbounded, so the REC inequality holds
along the subsequence.
SUBEXPONENTIAL case: limsup M(n)^(1/n) <= 1.
Then no lambda > 1 can work with linear growth. Instead, along
the records, M(N-l) / M(N) is subexponential in l: for every
lambda > 1, M(N-l)/M(N) <= lambda^(-l) holds only for l up
to some l_max that is sublinear in N. We will use the weaker
fact directly: for every fixed tau > 0, M(N-l) <= a^(tau*l) M(N)
for large N and all 0 <= l <= N. This is the only quantitative
input we carry into the subexponential case.
Both cases share the CONSTRUCTION below: given a large M(N), we
extract a nonconstant scalar function u and a finite measure mu.
1. LOCALIZATION TO A COORDINATE RECTANGLE
-----------------------------------------
The modulus M(N) is spread over the whole Z. We localize: it must
concentrate on SOME rectangle, else the sum argument fails.
LEMMA (LOCALIZATION). There is a coordinate chart Q = [0,1]^2 of Z,
with Qhat = [0,1] x [-1,2] inside the chart, such that (along a
subsequence of the record indices) the modulus
Mod_2 (F^{->}_Q, T_N) >= c * M(N),
where F^{->}_Q is the family of left-to-right path crossings of Q,
and c > 0 is independent of N.
PROOF. Cover Z by finitely many coordinate boxes [-1,1]^2 (this is
finite because Z is compact, and a chart is a homeomorphism to R^2
locally with a bounded-diameter coordinate patch). Each box sits in
a chart containing a larger box [-8,8]^2 of diameter less than
d_0 (the small minimum path diameter).
Every macroscopic path (diameter >= d_0) meets an inner box and
must leave the larger box through one of its four sides. Follow it
to its first meeting with the boundary of [-2,2]^2. If it exits
through the side x = +2, its segment after the last preceding
meeting with x = +1 crosses the strip [1,2] x [-2,2] left-to-right.
The other exit sides similarly give paths crossing analogous strips.
So there are finitely many STRIP-crossing families F_1, ..., F_J such
that every macroscopic path contains a member of one of them. If
weights w are admissible for the whole F_0 (macroscopic paths),
then they are in particular admissible for each F_j (because F_j
subset F_0 in terms of paths). Summing the squared weights:
M(N) = Mod_2(F_0, T_N) <= sum_j Mod_2(F_j, T_N)
<= J * max_j Mod_2(F_j, T_N).
One strip realizes the maximum, on a subsequence of the records.
Affine coordinates on that strip give the asserted Q with modulus
>= M(N)/J. []
The rectangle Q = [0,1]^2 has:
- TOP edge y = 0,
- BOTTOM edge y = 1,
- LEFT and RIGHT lateral edges x = 0 and x = 1,
- TOP BAND [0,1] x [-1,0] (above Q),
- BOTTOM BAND [0,1] x [1,2] (below Q).
The coordinates are only used for interpolation and topology;
all distances are the original visual metric d.
2. PLANAR BLOCKERS
------------------
We translate the classical fact "a barrier in a rectangle blocks all
top-to-bottom paths iff it contains a left-to-right continuum".
LEMMA (PLANAR BLOCKER). In a closed coordinate rectangle R:
(i) Every continuum (connected compact set) meeting both lateral
edges meets every top-to-bottom path.
(ii) Every compact set meeting all top-to-bottom paths contains a
continuum meeting both lateral edges.
PROOF of (i). Suppose C (lateral continuum) and pi (top-to-bottom
path) are disjoint. Enlarge R slightly, extending C to the left and
right boundaries of the enlarged rectangle and pi to the top and
bottom boundaries. The extensions remain disjoint because a
horizontal and vertical segment from a partial extension can only
meet at an original corner of C and pi, which are already disjoint.
Approximate pi within its positive clearance from C by a polygonal
path. After subdividing pi at intersections with the horizontal line
y = (top) + (bottom)/2 (or with any transverse cut), pi contains a
simple polygonal arc from top to bottom. The Jordan curve theorem
(applied to the closed curve formed by that arc plus the bottom
edge or suitable closure) says this arc separates the two lateral
edges. A connected set containing a point on each lateral edge must
meet the arc. Contradiction with disjointness of C and pi.
PROOF of (ii). Subdivide the rectangle into an ever-finer square
grid. Mark each closed square that meets the blocker K. No
edge-adjacent chain of UNMARKED squares joins top to bottom (else
those unmarked squares would trace a top-to-bottom path avoiding K,
contradiction with the blocker property).
Label A = the unmarked squares reachable from the top edge through
unmarked edge-adjacent squares. Append a top row of A-squares.
Label all remaining squares plus an appended bottom row B.
Draw the interfaces: edges between squares of different labels.
Interior vertices have even degree (they have an even number of A/B
changes around them). Each lateral boundary has an ODD number of ends
(the labels starting from the top go A ... A B ... B, which gives
an odd number of transitions). There are no top or bottom ends
(because the appended rows are uniformly A at top and B at bottom).
By a standard parity argument, some interface component joins the
lateral boundaries. (Otherwise every component contributes an even
number of ends to the only lateral boundary it meets.)
Along this interface, adjacent B squares are original MARKED squares
(by the construction of A as reachable-unmarked and B as everything
else). Also, an adjacent unmarked square on the A-side would have
been reachable (so it is not adjacent to the interface), and an
interface touch with the appended bottom row would give an unmarked
top-to-bottom chain (forbidden). So the B squares are indeed original
marked squares in K.
The union of these B squares is a connected compact set that meets
both lateral edges (the interface component does). Taking a
Hausdorff limit as the mesh tends to zero gives a continuum
contained in K and meeting both lateral edges. []
3. CROSSING DUALITY
-------------------
From the modulus bound on left-to-right crossings we produce
transverse weights that are admissible for top-to-bottom crossings
with tiny squared sum.
LEMMA (CROSSING DUAL). For each retained record N there are
nonnegative weights rho_q, q in T_N, admissible for top-to-bottom
crossings of Q, with
sum_q rho_q^2 <= C / M(N). (DUAL)
PROOF. Let V be the finite set of incidence vectors of left-to-right
crossings of Q (a vector v in {0,1}^{#T_N} for each crossing:
v_q = 1 if the crossing meets the cell B(q, 2r_N), else 0). Let P
be the point in the convex hull of V minimizing Euclidean norm.
First, P has nonnegative coordinates: it is a convex combination of
{0,1}-vectors. Second, its coordinates sum to at least one: every
left-to-right crossing meets at least one cell (it is a path of
positive length), so every v has sum >= 1.
Minimality property: for every v in V, the projection of v onto P is
minimized at the origin in the convex hull shifted by -P. Thus
langle P, v-P rangle >= 0, i.e. langle P, v rangle >= langle P, P
rangle = ||P||^2. Therefore P / ||P||^2 is admissible for
left-to-right crossings: langle P/||P||^2, v rangle >= 1.
Conversely, for any admissible weight w for left-to-right crossings,
langle w, P rangle >= 1 (since P is a convex combination of incidence
vectors v, each of which has langle w, v rangle >= 1). Combined with
weak duality, this gives Mod_2(F^{->}_Q, T_N) = 1/||P||^2.
Now, from the planar blocker lemma (i), every TOP-TO-BOTTOM
incidence vector u has inner product at least one with every
left-to-right crossing vector v (because any top-to-bottom path
must meet the continuum joining left to right, which is itself a
montage of cells meeting all the v_q; more carefully: a top-to-bottom
path and a left-to-right path must intersect, hence they share
at least one cell). Therefore langle u, P rangle >= 1 for all
top-to-bottom incidence vectors u.
Set rho = P. Then rho is admissible for top-to-bottom crossings.
Its squared sum is ||P||^2. Applying localization (Mod_2(F^{->}_Q)
>= cM(N)) gives ||P||^2 = 1/Mod_2(F^{->}_Q) <= C/M(N). []
4. CONTINUOUS FUNCTIONS FROM THE DUAL WEIGHTS
---------------------------------------------
Given the dual weights rho, we define a continuous function on Qhat
whose level sets separate. The raw definition is a "weighted distance
from the top edge" but continuity at the interpolation mesh is
subtle.
For a subset K of Qhat and a path pi subset Qhat, write
L_rho(pi) = sum_{q : B(q, 2r_N) intersect pi nonempty} rho_q.
For x in Q, define the raw value
b_N(x) = inf_{paths pi from top edge to x, contained in Q} L_rho(pi).
Then set the raw SCALAR
U^0_N(x) = min{ 1, b_N(x) } (raw scalar).
Extend: U^0_N is defined to be 0 on the top edge (b_N = 0 there),
and to be 1 on the bottom edge (admissibility says every top-to-bottom
crossing has L_rho at least 1, so the max cost on the bottom edge is
at least 1).
Key inequality: for any two points x, y in Qhat connected by a path
pi subset Qhat, we have
U^0_N(x) - U^0_N(y) <= L_rho(pi) and symmetrically, (RAW)
so |U^0_N(x) - U^0_N(y)| <= L_rho(pi) along any connecting path.
Reason: inside Q, a path from top to x plus pi itself plus a path
from y to top gives a top-to-x-plus-y-to-top monodromy, which can be
clipped at 1; this bounds one of U^0_N(x)-U^0_N(y) by L_rho(pi), and
symmetry the other.
Interpolate: triangulate Qhat by a finite coordinate grid containing
the band boundaries with every triangle of d-diameter less than r_N.
Let U^0_N be linear on each triangle. This gives a genuine
CONTINUOUS function Qhat -> [0,1], constant on the top and bottom
bands (by definition). Call this interpolant again U^0_N.
The interpolation just smooths out a piecewise-defined function; the
inequality (RAW) survives (with a small addition of r_N * rho_q for
cells near the interpolation edges) since r_N-small.
5. REGULARIZATION: SEPARATING LEVELS
------------------------------------
A continuous function can have level sets that are badly behaved
(disconnected, fractal). We need level sets that CROSS laterally.
The two path envelope trick (Cannon-Floyd-Parry style regularization)
gives this.
Define (for x in Qhat):
F_N(x) = inf_{pi from x to top edge, in Qhat} max_{z in pi} U^0_N(z).
H_N(x) = sup_{pi from x to bottom edge, in Qhat} inf_{z in pi} F_N(z).
LEMMA (LEVEL CONTINUA). The following hold:
(i) F_N, H_N are continuous on Qhat.
(ii) F_N >= U^0_N (with equality on the top band) and
H_N <= F_N (with equality on the bottom band).
(iii) For every path K subset Qhat:
osc_K H_N <= osc_K F_N <= osc_K U^0_N. (OSC)
(iv) For every 0 < t < 1, the set { H_N = t } intersect Q
contains a lateral-crossing continuum C_{N,t} of diameter at
least a fixed d_1 > 0.
(v) Every path in Qhat whose endpoint values STRICTLY straddle
t meets C_{N,t}.
PROOF. The setup F_N, H_N is a pair of envelope functions.
(i) Continuity: The envelopes are LIPSCHITZ UP TO CONTROL by U^0_N.
Since U^0_N is continuous on a compact Qhat, for any x, y close,
use a connecting path of the shortest possible form (a tiny
coordinate segment) to control F_N(x) - F_N(y) by oscillation of
U^0_N on that small segment. Similarly for H_N.
(ii) F_N >= U^0_N: a path from x to top always includes x itself,
so max_{z in pi} U^0_N(z) >= U^0_N(x). Taking inf over pi gives
F_N(x) >= U^0_N(x). For x on the top band, U^0_N(x) = 0, so
U^0_N(x) = 0 and F_N(x) = 0 (constant path in top).
H_N <= F_N: use a path from x to bottom, minimizing F_N on it,
gives H_N(x) <= inf F_N(pi) <= F_N(x). Similarly.
(iii) The parenthetical (OSC): we need a concatenation argument.
For x, y in K, take any path pi from x to top. Then pi
concatenated with any path from top to y (going down and into K)
gives a path from x to y through K extended by "top excursions".
The MAX along this concatenation is at least max{max_K U^0_N,
F_N(y) - epsilon_for_epsilon}. Taking inf over pi gives
F_N(x) <= max{ max_K U^0_N, F_N(y) }.
Suppose F_N(x) > F_N(y). Then max_K U^0_N >= F_N(x) (else the
right side had F_N(y), contradiction). Similarly F_N(y) bounds.
Both F_N(x) and F_N(y) lie in [min_K U^0_N, max_K U^0_N].
This gives osc_K F_N <= osc_K U^0_N.
Similarly H_N(y) >= min{ inf_K F_N, H_N(x) } (dual argument
using a path from x to bottom and a competitor from y). If
H_N(x) > H_N(y), then inf_K F_N <= H_N(x) (else worse), and
both H_N values lie in [inf_K F_N, sup_K F_N].
Combining osc_K F_N <= osc_K U^0_N and H_N's contribution via
F_N's bounds gives (OSC).
(iv) The set {H_N = t} INTERSECT Q is compact (H_N is continuous, Q
is compact) and blocks every top-to-bottom path in Q: any
top-to-bottom path has values going from H_N = 0 at top to
H_N = 1 at bottom, so it hits t somewhere. Thus {H_N = t} is a
compact set blocking top-to-bottom paths in Q. By the planar
blocker lemma (ii), it contains a lateral continuum C_{N,t}.
Take d_1 = the distance between the two lateral edges of Q
(positive constant). Diameter of C_{N,t} is at least d_1.
(v) Given a path with strictly straddling endpoints t_low < t <
t_high, connect the low part to the top band within the strict
sublevel { H_N < t }, and the high part to the bottom band within
the strict superlevel { H_N > t }. The completed path goes
top-to-bottom and thus meets C_{N,t}. The added pieces avoid the
level set {H_N = t}, so the meeting happens on the original
path's segment near the level (or on the strict-side extensions
that are common). Either way, the ORIGINAL path must meet C_{N,t}.
The details of (v): we have three regions above (top band where
H_N = 0), below (bottom band where H_N = 1), and the strict
sublevel {H_N < t} / strict superlevel {H_N > t}. The path with
endpoint values straddling t must enter and leave the strict sides,
which we showed join their bands without meeting the level. So the
completed path crosses all levels including t. The added pieces
avoid t by construction, so the crossing point is on the interior
of the completed path, which is contained in the original path (up to
band extensions). Combined with the planar blocker lemma again, the
original path's interior (before extension) meets C_{N,t}. []
Figure PART V shows a schematic: the blue curve is C_{N,t}, the
ochre dashed pieces are the strict-side extensions that complete an
original path eta into a top-to-bottom crossing.
6. LOCAL OSCILLATION
--------------------
We can now bound local oscillation of H_N by nearby rho-values.
LEMMA (LOCAL OSCILLATION). For fixed compact K subset U =
int_Z(Qhat) and J > 0, uniformly for x in K and sufficiently large N:
osc_{Bbar(x, Jr_N)} H_N
<= sum_{q: d(q,x) <= C(J) r_N} rho_q. (LOC-OSC)
Here C(J) depends only on J and the additive constants.
Consequently, for any fixed group element g and compact K' subset gU
with J >= 2:
E_N^(J)(H_N o g^(-1); K') <= C(g, K', J) / M(N) -> 0. (SMALL-E)
The localization energy E_N^(J) is defined in (ENRG) below; the main
point is that squared oscillation sums tend to zero.
PROOF. Take x, y in Bbar(x, Jr_N), and join them by a path pi subset
B(x, LJr_N) via the paths tool. Then for all cells meeting pi, the
Gromov product (or the raw-score function) bounds their U^0_N value
difference by sum_q in cells-meeting-pi of rho_q. The raw inequality
(RAW) extends to the interpolant up to r_N-small errors, hence bounds
H_N-oscillation on the original endpoints by the same sum.
Specifically: the triangle vertices meeting pi come from
cells with centers in B(x, (LJ+1) r_N). Any two such vertices join
inside B(x, L(LJ+1) r_N) subset U for large N. Applying (RAW),
then the level-continua inequality (OSC)
osc_K H_N <= osc_K U^0_N,
gives (LOC-OSC) with C(J) = L(LJ+1) + 3.
For SMALL-E, use packing: summing squared sums of localized rho over
K' charges boundedly many weights multiple times. Use bi-Lipschitz
of g to relate target ball preimages to source balls. All fits inside
U. Combined with DUAL (sum rho_q^2 <= C/M(N)), SMALL-E follows. []
7. ANNULAR ESTIMATE
-------------------
The final estimate connects local oscillation of H_N at scale R to
the surrounding DUAL energy at scale r_N, using expansions.
LEMMA (ANNULAR ESTIMATE). For every compact K subset U, there are
R_* > 0 and integer m_0 such that, for x in K, R = r_l <= R_*, and
N - l >= m_0, any two points joined by a path in B(x, R/2) have
H_N-value difference DELTA satisfying
DELTA^2 <= C M(N-l) * sum_{q: R/4 <= d(q,x) <= 4R} rho_q^2.
(ANN)
PROOF. Choose R_* with 4 R_* < d_1 and all closed 5R_*-balls about
K contained in U. All following constraints on m_0 are on
r_N / R = a^(-(N-l)), positive powers of a.
Expand at x, l by h = v^(-1) where v has depth l on a ray to x.
Choose admissible weights omega at scale N-l with squared sum
M(N-l) (from the definition of M). Define aggregate weights:
A = { q in T_N : R/2 <= d(q,x) <= 3R }
sigma_q = sum_{xi in T_{N-l} : d(xi, hq) <= C_0 r_{N-l}} omega_xi
for q in A, where C_0 > C_E + 2.
Then expansion on B(x, 3R) separates the images hq by a fixed
multiple of r_{N-l}. Packing bounds the overlap in both directions.
Direct computation:
sum_{q in A} sigma_q^2 <= C M(N-l). (AGG-ENERGY)
Now take two points y1, y2 joined by a path in B(x, R/2), and let I
be the open interval between their H_N-values. For each t in I, the
level continuum C_{N,t} meets the given path, and since
diam(C_{N,t}) >= d_1 > 4R, C_{N,t} must EXIT B(x, 2R). By the paths
tool, join two such points of C_{N,t} inside its r_N/4-neighborhood,
and retain a radial subpath eta_t between radii R and 2R. Then
eta_t has diameter at least R, so h eta_t has diameter at least
R / (2R) = 1/2 > c, hence is MACROSCOPIC (diameter >= d_0) by the
choice of d_0 in the localization lemma.
If a target cell B(xi, 2 r_{N-l}) meets h(eta_t) at some hy, choose
q in T_N with d(q,y) <= r_N. For large m_0, this center q falls in A,
and d(hq, hy) <= C_E r_N / R, so omega_xi (which is charged to hq)
appears in sigma_q. Some level point of C_{N,t} lies within r_N/4 of
y, hence within Bbar(q, 2r_N). Therefore t lies in the closed range
interval I_q of H_N on Bbar(q, 2r_N).
Accounting for all target cells gives, for t in I:
1 <= sum_{q in A} sigma_q * 1_{t in I_q}. (COVER)
Integrate over t in I:
|I| <= sum_{q in A} sigma_q * |I_q|.
Now use |I_q| >= osc_{B(q, 2r_N)} H_N. But we already have (LOC-OSC)
giving osc bounds. Specifically, by Cauchy-Schwarz:
|I|^2 <= (sum_{q in A} sigma_q^2) * (sum_{q in A} |I_q|^2).
The sum of squared intervallengths |I_q|^2 is bounded by oscillation
squared, which by LOCAL OSCILLATION (LOC-OSC) with J=2 is (up to
bounded overlap summation) bounded by the localized rho-energy.
Combining with AGG-ENERGY gives (ANN). Take C(2) r_N < R/4 to
confine every charged center to R/4 <= d(q,x) <= 4R. []
8. THE SCALAR LIMIT u AND MEASURE mu
------------------------------------
Now we combine annular estimates with records (REC) to extract u and
mu.
Define the finite measures
mu_N = M(N) * sum_{q in T_N} rho_q^2 * delta_q (on Z)
(the Dirac mass delta_q at the cell center q, scaled by M(N) times
squared weight). Then
mu_N(Z) = M(N) * sum_q rho_q^2 <= C (MEAS-MASS)
by DUAL. So mu_N has uniformly bounded total mass, and by weak
compactness of the space of measures on the compact Z, a
subsequence converges to a finite mu.
PROPOSITION (SCALAR LIMIT). Along a subsequence of the record indices,
H_N -> u locally uniformly on U = int_Z(Qhat), and mu_N -> mu weakly
(on Z), where u : U -> [0,1] is continuous and nonconstant, mu is
finite with bounded mass <= C, u = 0 on the top band interior and
u = 1 on the bottom band interior, and for every compact K subset U,
sufficiently small s > 0, and uniformly in x in K:
(osc_{Bbar(x, s)} u)^2 <= C * mu(Bbar(x, C_mu * s)). (MEAS-OSC)
If lambda > 1, u is locally HOLDER CONTINUOUS on U.
PROOF. Step A: local equicontinuity of H_N.
Fix compact K subset U and x, y in K with d(x,y) <= r_k. Join x to y
inside B(x, L r_k) by the paths tool. Choose b with a^b > 2L. Consider
indices l <= k - b in an arithmetic progression of step b_1 with
a^(b_1) > 16. The annuli {r_l/4 <= d(x,cdot) <= 4r_l} are pairwise
disjoint (by choice of step).
For N large relative to k (so that the ANN estimate applies), the
ANN estimate applied at x and l gives:
|H_N(x) - H_N(y)|^2 <= C M(N-l) * sum_{q in A_l(x)} rho_q^2
= C lambda^(-l) * mu_N (A_l(x)) (ANN-AS-DELTA)
where A_l(x) = { r_l/4 <= d(x,cdot) <= 4r_l } and we used (REC):
M(N-l) <= lambda^(-l) M(N).
Sum over the at least ck disjoint annuli:
|H_N(x) - H_N(y)|^2 <= C lambda^(-l_min) * mu_N(Z) / 1
<= C_K * k^(-1) for the worst case; more
carefully with lambda > 1 we can choose l freely, giving
the best bound C_K k^(-1) (general case, lambda = 1 in the
subexponential case where the sum over many annuli is used).
Precisely:
- in the EXPONENTIAL case (lambda > 1), take l = k - b, giving
|H_N(x) - H_N(y)|^2 <= C_K lambda^(-k).
- In the general case, sum over at least ck disjoint annuli,
giving |H_N(x) - H_N(y)|^2 <= C_K / k.
Either way, H_N is EQUICONTINUOUS on K, uniformly in N.
Step B: extraction of a subsequence.
For each tolerance, we have a tail-uniform control; the finitely many
omitted functions are individually uniformly continuous. So H_N is
equicontinuous on K. By the Arzela-Ascoli theorem on a compact
exhaustion of U, there is a uniformly convergent subsequence on each
compact subset. Diagonalize to get a single subsequence with
H_N -> u locally uniformly on U.
Then mu_N has bounded total mass, so by the Banach-Alaoglu theorem
applied to finite measures on the compact Z, a further subsequence
converges weakly to a finite mu.
Step C: nonconstancy.
The two band values give u = 0 on the top band and u = 1 on the
bottom band. Since U contains the interiors of both bands, u is
nonconstant.
Step D: measure estimate.
Fix A > 2L. For small s, choose R = r_l in [As, aAs). Any two
points of Bbar(x, s) join in B(x, Ls) subset B(x, R/2). The annular
estimate gives
(osc_{Bbar(x, s)} H_N)^2 <= C * mu_N(Bbar(x, 4 a A s)).
Local uniform convergence passes the left side to u; the weak
closed-set inequality (limsup of mu_N of closed set <= mu of closed
set) passes the right to mu. Take C_mu = 4 a A.
Step E: Holder (lambda > 1).
For lambda > 1, the single-scale bound gives
|H_N(x) - H_N(y)|^2 <= C_K lambda^(-k)
where k is such that d(x,y) = r_k, i.e., the local Holder exponent is
at least (1/2) log(lambda)/log(a). This passes to u. []
9. LEVEL CONTINUA IN THE LIMIT
------------------------------
For each t in (0,1), the level continua C_{N,t} of H_N are subsets
of the compact Q. Take a Hausdorff limit C_t subset Q along a
subsequence (which may depend on t). Then C_t is compact and
connected (limit of continua), has diameter at least d_1, and by
local uniformity of H_N -> u, u = t on C_t intersect U.
A fixed compact path with endpoint values strictly straddling t
also straddles t for H_N for every sufficiently large N, so meets
those C_{N,t}; taking the limit, it meets C_t. Thus:
For each t in (0,1) there is a continuum C_t subset Q of diameter
at least d_1, with u = t on C_t intersect U, meeting every compact
path in U whose endpoint values strictly straddle t. (LEVEL-U)
The subsequence for C_t may depend on t; no simultaneous selection is
needed at this stage.
10. FINITE-CELL LEVEL COMPARISON
---------------------------------
The next tool is a general fact about oscillation energies sweeping
level continua.
LEMMA (CROSS-ENERGY). Let f, w be continuous functions on an open
V subset Z, let K subset V be compact, and let I subset R be an
interval. Suppose for every t in I there is a continuum D_t subset K
with f = t on D_t and osc_{D_t} w >= b_0 > 0. Then, for sufficiently
large n:
b_0 |I| <= sum_{q : B(q, 2r_n) intersect K nonempty}
osc_{B(q, 2r_n)} f * osc_{B(q, 2r_n)} w. (CROSS)
PROOF. Let I_q(f) be the closed range interval of f on the closed
cell Bbar(q, 2r_n). For large n, all these cells lie in V (since
K subset V).
For any t in I, the continuous image w(D_t) is an interval of length
>= b_0. Cover w(D_t) by the range intervals I_q(w) of cells meeting D_t:
this is possible because the union of the B(q, 2r_n) with D_t subset
covers D_t, so w(D_t) is contained in union of w(B(q, 2r_n)).
Each cell meeting D_t has t in I_q(f) (since f = t on D_t, and the
cell contains a point of D_t where f = t). Thus integrating indicator
functions:
b_0 <= sum_q osc_{cell_q} w * 1_{I_q(f) contains t}. (PTW)
Integrate over t in I:
b_0 * |I| <= sum_q osc_{cell_q} w * |I ∩ I_q(f)|
<= sum_q osc_{cell_q} w * osc_{cell_q} f. (INT)
Note only finitely many interval indicators are integrated (one per
cell), so no measurable selection of D_t is needed. []
11. STOPPING A CONTINUUM AT A SPHERE
------------------------------------
The last tool: continuum slicing.
LEMMA (STOP-CONTINUUM). Let D be a continuum in Z, p in Z, r > 0. If
x in D minus Bbar(p, r) and D intersects B(p, r) (open ball), then D
contains a subcontinuum K with x in K, K meeting the sphere
{y : d(y,p) = r}, and K subset of D minus B(p, r).
PROOF. In a compact metric space X, the COMPONENT of a point x is
the intersection Q_x of its clopen neighborhoods.
Lemma (classical): the component Q_x is connected.
Proof of connectedness: suppose Q_x = A union B with A, B compact
and nonempty in disjoint open O_A, O_B. The remainder X minus (O_A
union O_B) is compact; its distance from Q_x is positive, so there is
a finite intersection W of clopen neighborhoods of x with
W subset O_A union O_B. Then W ∩ O_A (where x lies in O_A) is both
open and closed in X and misses Q_x ∩ B. Contradiction with Q_x
being the intersection.
Same argument shows that every open neighborhood of Q_x contains a
clopen neighborhood of Q_x (in X).
Apply this to X = D minus B(p, r) (which is compact, x lies in it),
and consider the component C of x in X.
Claim: C meets the boundary sphere d(y,p) = r.
Proof of claim: Suppose C misses the sphere. Then C subset of
D minus Bbar(p, r), which is open in D minus B(p, r). Find a clopen
neighborhood W of C in X with W subset D minus Bbar(p, r) (possible
by the class lemma: any open neighborhood contains a clopen one).
Then W is clopen in D minus B(p, r) AND in X.
But W is separated from D intersect Bbar(p,r) (by definition of
W subset D minus Bbar(p, r)). D minus Bbar(p, r) is relatively open
in D and D is connected. So D cannot split into W (open in D minus
B(p, r), which projects to open in D minus Bbar(p, r)) and the rest.
Contradiction. So the claim holds.
Thus C is the required subcontinuum: through x, meeting the sphere,
and inside D minus B(p, r). []
12. WHAT IS READY
-----------------
We now have, given the unbounded M(N), a nonconstant u : U -> [0,1],
a finite mu on Z with mass bounded by C (independent of N), and:
(MEAS-OSC) (osc_{B(x,s)} u)^2 <= C mu(B(x, C_mu s)), (mu-estimate)
(SMALL-E) local squared oscillation energies of FIXED transforms
of H_N tend to zero,
(LEVEL-U) level continua C_t exhibit u = t on C_t ∩ U,
(CROSS) a finite-cell averaging identity.
These tools now drive the contradiction in both the exponential and
subexponential cases. PART VI uses them to force a nonconstant function
on a punctured sphere and then a contradiction. PART VII uses them plus
a probability flow to force the measure comparison at the boundary of
Z.
================================================================================
PART VI. THE EXPONENTIAL CASE: LAMBDA > 1 -> CONTRADICTION
================================================================================
Assume now that the records are EXPONENTIAL: lambda > 1 in (REC).
Then u is locally Holder on U (PART V, Step E of Scalar Limit).
The idea: magnify u to a nonconstant function on a punctured sphere,
and force transforms with different punctures to be locally constant
wherever u varies. A second magnification then produces a
contradiction.
Throughout: all scalar-approximation indices N belong to the
subsequence of PART V's Scalar Limit proposition.
1. HOLDER MAGNIFICATION
-----------------------
The first blowup: take a small region where u varies, and zoom in
until the variation becomes O(1) while the diameter stays bounded
away from a puncture.
LEMMA (HOLDER MAGNIFICATION). Let f be a real-valued, locally
Holder continuous function on an open subset Omega of Z. Suppose
O is open with closure(O) subset Omega, and O contains a compact
path on which f is nonconstant. Fix z_0 in Z with distance(z_0, O)
at least epsilon_* > 0 (positivity assured). Then there exist depths
k_i -> infinity, centers x_i in O, positive numbers sigma_i, and
expansions h_i = v_i^{-1} at (x_i, k_i) such that (after subsequence):
(i) p_i = h_i(z_0) -> b in Z,
(ii) distance(x_i, Z \ O) / r_{k_i} -> infinity,
(iii) every compact subset of Z \ {b} eventually lies in h_i(Omega),
(iv) the functions
F_i(z) = ( f(h_i^{-1}(z)) - f(x_i) ) / sigma_i (BLOWUP)
converge locally uniformly on Z \ {b} to a nonconstant locally
Holder function F.
PROOF. Write delta(x) = distance(x, Z \ O). Fix a magnifying power
eta > Q (where Q is the packing exponent from PART IV).
Holder: since O's closure is compact inside Omega and f is locally
Holder on Omega, finite cover of closure(O) by Holder balls gives a
UNIFORM Holder bound with exponent alpha_0 > 0 on a neighborhood of
closure(O).
For k large, define the supremum
A_k = sup_{ x in O } delta(x)^eta * osc_{Bbar(x, r_k)} f. (SUP-F)
Because B(x, r_k) subset Omega (since delta(x) >= r_k for small r_k,
because closure(O) is compact in Omega), and the Holder bound applies,
A_k <= C_holder * a^(-alpha_0 k). (UPPER-A)
LOWER BOUND on A_k: cover a compact variation path P subset O by at
most C r_k^{-Q} radius-r_k balls centered on P (packing). Their range
intervals cover f(P), which is an interval of positive length (f is
nonconstant on P). So one ball has oscillation at least c r_k^Q times
the interval length, giving:
A_k >= c a^(-Q k) * (min_P delta)^eta. (LOWER-A)
Thus c a^{-Qk} <= A_k <= C a^{-alpha_0 k}. Fix 0 < alpha < alpha_0.
Then A_k * a^{alpha k} -> 0. Pick a subsequence of indices k_j
attaining the maximum of A_k * a^{alpha k} over the tail [k_j, infinity),
so that
A_{k+j} <= a^{-alpha j} * A_k for all j >= 0. (TAIL-DECAY)
Choose x_k in O attaining at least half of the supremum A_k.
Set delta_k = delta(x_k), and
sigma_k = A_k / delta_k^eta. (SIGMA)
Boundedness of f and LOWER-A give delta_k^eta >= c r_k^Q, hence
delta_k / r_k >= c r_k^{Q/eta - 1} -> infinity (since eta > Q).
Expand at (x_k, k) and subsequence with p_k = h_k(z_0) -> b in Z.
Preimage estimate for compact K subset Z \ {b}: by the puncture
estimate (E5), if d(z,b) >= 1/j for large k, the preimage
h_k^{-1}(z) lies in B(x_k, C_K r_k) subset O. This gives (iii) with
the diagonal selection below.
Holder of F_k: on every fixed multiple of B(x_k, r_k), delta is
comparable to delta_k. Thus by TAIL-DECAY:
|f(y) - f(y')| / sigma_k <= C_K a^{-alpha j}
whenever d(y,y') <= r_{k+j}, j >= 0. (LOCAL-HOLD)
For d(z,w) small, choose j >= 0 with a^{-j} >= C_K d(z,w) > a^{-j-1}.
Then a uniform LOCAL Holder estimate for F_k on K follows from
(LOCAL-HOLD).
Boundedness: paths tool joins h_k^{-1}(z) to x_k inside a fixed
multiple of B(x_k, r_k). Packing covers this path by bounded number
of radius-r_k balls centered on the path; each has oscillation
<= C sigma_k. So |F_k| <= C_K on K.
Arzela-Ascoli on compact exhaustion of Z \ {b} gives a locally
uniform, locally Holder limit.
Nonconstancy: osc of F_k on h_k(Bbar(x_k, r_k)) is at least 1/2
(because u changes by sigma_k = A_k / delta_k^eta on this scale, and
normalization makes this O(1)). The lower puncture bound puts these
image balls in a fixed compact set off b, so by local uniform
convergence, the limit F is nonconstant there. []
2. DIAGONAL APPROXIMANTS
------------------------
Now apply the Holder magnification to u on U (Omega = U), taking a
small variation region O subset U and z_0 outside closure(O). Get
limit W, puncture b, expansions h_j, normalization sigma_j.
We want to intertwine with the numeric scalar limit H_N. Since
H_N -> u is only locally uniform, we must "thin down to a common
subsequence" carefully.
LEMMA (DIAGONAL APPROXIMANTS). After thinning the expansions, there
are increasing scalar indices N_j such that, with
f_j(z) = ( H_{N_j}(h_j^{-1}(z)) - u(x_j) ) / sigma_j,
for z in h_j(Qhat), (F-J)
convergence f_j -> W is locally uniform on Z \ {b}. For every FIXED
group element g, compact K subset Z \ {g b}, and fixed J >= 2,
E_{N_j}^(J) (f_j o g^{-1}; K) -> 0. (ENERGY-J)
All indicated enlarged balls are eventually contained in the
functions' domains.
If a compact path P subset Z \ {b} has endpoint values of W strictly
straddling a compact positive-length interval I, then, for all large
j and every t in I, there is a continuum D_{j,t} in h_j(Qhat) with
f_j = t on D_{j,t}, (D-LEVEL)
D_{j,t} intersect P is nonempty, (D-MEET)
sup_{t in I} distance(b, D_{j,t}) -> 0. (D-NEAR)
PROOF. For j >= 4, put K_j = { z : d(z, b) >= 1/j } and
L_j = { z : d(z, b) >= 1/(2j) }.
CHOOSE the jth expansion with h_j^{-1}(L_j) subset U and its
normalized u-transform within 1/j of W on L_j. HOLD THIS EXPANSION
AND ITS NORMALIZATION FIXED.
Then choose N_j > N_{j-1} so large that three things hold
simultaneously:
sup_{z in L_j} |H_{N_j}(h_j^{-1}z) - u(h_j^{-1}z)| / sigma_j <= 1/j,
(SLOW-CONVERGE)
E_{N_j}^(j) (f_j; K_j) <= 1/j, (SMALL-J)
(j + 2) * r_{N_j} < 1/(2j). (SMALL-MESH)
The uniform approximation (SLOW-CONVERGE) follows from PART V
(Scalar Limit). With h_j and sigma_j now fixed, PART V's local
oscillation lemma gives
E_{N_j}^(j)(f_j; K_j) <= C(h_j, K_j, j) / (sigma_j^2 M(N_j)),
which can be made < 1/j by increasing N_j because M(N_j) -> infinity
along records. (SMALL-MESH) keeps every enlarged ball in L_j.
Then f_j -> W locally uniformly (uniform approximation + already-
established u-transform convergence on L_j), and the energy bound
(ENERGY-J) holds at every fixed enlargement off b.
For a FIXED g (or g_j = identity) — actually the general case: fix
g in G, and compact K subset Z \ {gb}. Pick a compact neighborhood
K' of g^{-1}K avoiding b. Pulling back each target ball and
choosing a nearest source net point gives, by bi-Lipschitz
distortion and packing,
E_n^(J)(F o g^{-1}; K) <= C_g * E_n^(J')(F; K') (TRANSFER)
for some fixed J' >= 2 and all large n. Each source center receives
only boundedly many target centers. Eventually K' subset K_j and
J' <= j, giving ENERGY-J.
Uniform convergence makes endpoint values of f_j straddle I for
large j. The source level tau_j(t) = u(x_j) + sigma_j * t lies in
(0,1) for large j (since u(x_j) in (0,1) for both band-interior
reasons). The PART V level continuum C_{N_j, tau_j(t)} gives us
D_{j,t} = h_j(C_{N_j, tau_j(t)}), which:
- lies on level f_j = t (by choice of source level),
- meets P (because f_j -> W on P and the level straddles W-
endpoints on P, so the scalar level continuum crosses P),
- has source diameter >= d_1 (so its source contains a point at
distance >= d_1/2 from x_j); by the puncture estimate,
distance(b, D_{j,t}) <= d(b, p_j) + C r_{k_j} -> 0 uniformly
in t. []
3. TRANSFORM COMPARISON
-----------------------
The next step: force each transform with a different puncture to be
CONSTANT wherever W is not locally constant. This is an analytic
rigidity statement.
First, notation: define N(F) = set of points in the domain of a
continuous function F where F is NOT locally constant.
LEMMA (TRANSFORM COMPARISON). Suppose c = g(b) is not equal to b,
and V = W o g^{-1}. Then
V(x) = V(b) for every x in N(W) \ {c}. (TRANSFORM-LEVEL)
More generally, if W_s = W o s^{-1}, p_s = s b, and p_s != p_t:
W_t(x) = W_t(p_s) for every x in N(W_s) \ {p_t}.
(TWO-LEVEL)
PROOF. Suppose x in N(W) \ {c} and V(x) != V(b). Pick epsilon > 0
and a coordinate disk about x, compactly contained off b and c, on
which |V - V(b)| >= 4 epsilon. This disk contains a compact variation
path P for W (because x in N(W)).
Choose a compact positive-length interval I strictly between the
endpoint values of W on P (this is possible because P is a variation
path), avoiding W(c) (since W is continuous at c, small change).
Continuity: there is epsilon_c > 0 such that Bbar(c, epsilon_c)
avoids b and its W-range has positive distance from I. Consequently,
for all large j, the continua D_{j,t} (from the diagonal approximants
lemma) AVOID this closed ball around c (because f_j -> W uniformly on
the compact region, and W-range near c stays away from I).
Now choose rho > 0 so that Bbar(b, rho) avoids c, P, and satisfies
|V - V(b)| <= epsilon THERE (this is continuity of V at b). For
w_j = f_j o g^{-1}: local uniform convergence gives, for all large j:
|w_j - V(b)| >= 3 eps on P, (HIGH-P)
|w_j - V(b)| <= 2 eps on Bbar(b, rho). (LOW-BALL)
Each D_{j,t} meets P (D-MEET) and enters B(b, rho) (D-NEAR).
By the stop-continuum lemma (PART V.11) we can extract a subcontinuum
joining a chosen point of P (where w_j - V(b) is large) to the sphere
of radius rho (where w_j - V(b) is small), contained in
K = { z : d(z, b) >= rho, d(z, c) >= epsilon_c }. (K-FRAME)
This subcontinuum:
- lies on the level f_j = t (D-LEVEL),
- has w_j-oscillation at least epsilon (by HIGH-P and LOW-BALL,
since both endpoint regions are within the subcontinuum).
A fixed compact neighborhood of K avoiding both punctures EVENTUALLY
lies in h_j(U) intersect g h_j(U) (both expansions cover it), and
contains every closed radius-2r_{N_j} cell meeting K.
Now apply the finite-cell CROSS-ENERGY (PART V.10) with
f = f_j, w = w_j, level family D_t = the subcontinua (on level t),
and I the chosen interval.
eps |I| <= sum_q osc(cell_q) f_j * osc(cell_q) w_j.
(CROSS-USE)
By Cauchy-Schwarz:
(eps |I|)^2 <= E^(2)(f_j; K) * E^(2)(w_j; K).
Both energies tend to ZERO by ENERGY-J (with g = identity for the
first, with g = the chosen element for the second). Contradiction
with eps, |I| > 0 fixed. All sets and g are fixed in this limit.
Hence V(x) = V(b) for all x in N(W) \ {c}. This is (TRANSFORM-LEVEL).
For (TWO-LEVEL): apply (TRANSFORM-LEVEL) at s^{-1}(x) with the group
element s^{-1} t. Concretely, set W_s = W o s^{-1}, W_t = W o t^{-1}.
If p_s = s b != t b = p_t, then W_t is a transform of W by the group
element t s^{-1}, and its "puncture" is t b = p_t. Apply
(TRANSFORM-LEVEL) with b replaced by the s-perspective puncture, c
replaced by p_t: W_t(x) = W_t(p_s) for all x in N(W_s) \ {p_t}. []
4. STABILIZER DIRECTIONS
------------------------
We need a bound on the stabilizer of b in G, to control repeated
punctures in the second magnification.
LEMMA (STABILIZER). Let E = Stab_G(b). The vertices of E have at
most TWO accumulation directions on Z.
PROOF. Fix a based ray xi to b. For q in E, |q| <= t, thinness of
the ideal triangle with sides [e, q], xi, q xi puts q xi(n) uniformly
near xi when n is sufficiently larger than t. The nearby parameter
lies in the interval [n - t - C, n + t + C].
For one common integer n (large enough that q xi(n) is defined and
distinct for all distinct q in E with |q| <= t -- possible by
injectivity of the map q -> q xi(n) for distinct q), what matters is
that |q| is bounded: the number of possible q with |q| <= t is
bounded. Bounded valency of the Cayley graph gives
# { q in E : |q| <= t } <= C'(t + 1). (STAB-GROWTH)
Now suppose a second accumulation direction exists beyond b. Take
q_n -> x != b along a subsequence, and simultaneously q_n^{-1} -> y
along a subsequence. Base change:
(q_n | q_n z)_e = |q_n| - (q_n^{-1} | z)_e + O(1)
>= |q_n| - O(1) whenever z avoids y.
By the prefixes lemma, q_n z -> x uniformly on compact sets
avoiding y. But q_n b = b (by definition of E), so only y = b is
consistent with continuity. Contradiction with y != b.
Choose small disjoint neighborhoods of b and x, with a point outside
their union. For large n, q = q_n sends the complement of the first
neighborhood into the second. Its POSITIVE iterates never return the
outside point (never reach back to near b), so q has infinite order.
Such q is therefore LOXODROMIC (hyperbolic isometry of H^3, or its
boundary action), with endpoints b and x' (some other boundary point).
Now suppose some h in E MOVED x'. Then the cyclic subgroups generated
by q and h q h^{-1} would intersect trivially: a nonzero common power
would have to fix the same pair of endpoints {b, x'}, but q fixes
b and x', while h q h^{-1} fixes b and h(x'). If h(x') != x', the
only common fixed pair would be {b}, contradicting the translation
nature of loxodromics. Their m^2 products q^i (h q h^{-1})^j,
1 <= i,j <= m, would be DISTINCT and have length O(m), contradicting
STAB-GROWTH (which bounds the count by linear-in-m).
Thus every element of E fixes both endpoints b and x'. Its image of a
geodesic line between b and x' is another line at uniformly bounded
Hausdorff distance (any isometry of a delta-hyperbolic space that
fixes both ends stabilises the geodesic line between them up to
bounded error). So the orbit of e stays a fixed distance from that
line, and its only possible accumulation directions are the two
endpoints. []
5. ASSEMBLING THE CONTRADICTION
--------------------------------
Now the second magnification, using W from the first.
PROPOSITION (NO EXPONENTIAL RECORDS). The contradiction setup (PART V
assume M unbounded with lambda > 1) cannot hold.
PROOF. Let E = Stab(b), and Lambda_E its at-most-two accumulation
directions (from STABILIZER).
Choose a VARIATION REGION O for W whose closure avoids the finite set
{b} union Lambda_E. Such a region exists: otherwise W would be locally
constant near every point of {b} union Lambda_E, and by continuity of
W on Z \ {b} (which is connected as a sphere minus a point) plus
finite additivity, W would be globally constant on its domain;
contradiction with nonconstancy of W from HOLDER MAGNIFICATION.
Apply HOLDER MAGNIFICATION to W on Z \ {b} with this O and z_0 = b.
The expansions are s_i = v_i^{-1} with centers y_i in O, depths
l_i -> infinity, and punctures p_i = s_i b -> b_* (subsequence
convergence). Positive AFFINE normalizations of W_i = W o s_i^{-1}
converge locally uniformly (on Z \ {b_*}) to a continuous nonconstant
W_*.
CLAIM: The punctures p_i have a pairwise distinct subsequence.
Proof of claim. Suppose v_i^{-1} b = v^{-1} b infinitely often.
Then v_i v^{-1} in E (by definition of stabilizer). Take a
subsequence with y_i -> y in closure(O). Since v_i lies at depth l_i
on a ray to y_i (by the prefixes lemma), v_i -> y in boundary
direction (as an accumulation direction of the based rays, hence of
the vertices v_i). Bounded right multiplication preserves this
accumulation direction. So y = v_i's aggregation direction is in
Stab_E's accumulation set long the sequence. Since the accumulation
directions of E are given by STABILIZER (only two, both in
Lambda_E), we get y in Lambda_E. But O's closure avoids
Lambda_E. Contradiction. Claim proved.
Now choose TWO coordinate disks D_1, D_2 in Z \ {b_*}, with compact
closures off b_*, having DISJOINT W_*-range intervals, each
containing a compact variation path.
Construction of D_1, D_2:
- Follow a variation path of W_* (which exists because W_* is
nonconstant) to its first hitting points of two distinct
intermediate values alpha_1 < alpha_2 (both attained by W_*).
- Just before the first hitting time, W_* is not locally constant
at those points (the local variation is genuine).
- Continuity of W_* gives DISJOINT range disks D_1, D_2 around
those points (shrunk until disjoint range intervals).
For every sufficiently large i, the puncture p_i is OUTSIDE the
closed disks D_1, D_2 (since p_i -> b_* and D_1, D_2 have compact
closures off b_*).
Uniform convergence of the normalized W_i preserves both:
- the disjoint range intervals on D_1, D_2,
- the positive variation on the two fixed paths.
Since AFFINE normalization is bi-Lipschitz, these properties are
preserved. Hence N(W_i) MEETS each disk D_1, D_2.
Now fix two DISTINCT such indices i, j (both large). Apply
(TWO-LEVEL) with W_s = W_i, p_s = s_i^{-1} (b) = p_i^?, and view W_j
as a transform. The lemma (TWO-LEVEL) gives:
W_j(x) = W_j(p_i^?) for all x in N(W_i) \ {p_j^?}.
We need W_j(p_i) not W_j(p_i^?). Apply (TWO-LEVEL) at the
appropriate shift.
Concretely (matching the original proof): if two transforms
W_i = W o s_i^{-1}, W_j = W o s_j^{-1} have distinct punctures
p_i != p_j, then (TWO-LEVEL) says
W_j(x) = W_j(p_i) for all x in N(W_i) \ {p_j}. (ALIGN)
Now take ONE point x_1 in D_1 intersect N(W_i) and ONE point
x_2 in D_2 intersect N(W_i). Both are in N(W_i) (by choice), and
neither is p_j (both D_1, D_2 are off b_* and p_j near b_*).
By (ALIGN):
W_j(x_1) = W_j(p_i) = W_j(x_2).
But the range intervals of W_j on D_1 and D_2 are DISJOINT (from
uniform convergence of W_i -> W_* with disjoint W_*-range, and
W_j is close to W_* for large j).
Contradiction. No such W_j can exist. Hence no lambda > 1 setup
can hold. []
6. WHAT JUST HAPPENED
---------------------
The exponential case killed itself in three steps:
A. AMPLIFICATION. Holder regularity let us zoom u into a nonconstant
W on a punctured sphere. Amplification preserves the ENERGIES of
every fixed transform, allowing diagonal H_N to approximate W
uniformly.
B. RIGIDITY. Comparing transforms with distinct punctures, the
finite-cell energy comparison forces each transform to be
constant on the not-locally-constant set of W.
C. SECOND AMPLIFICATION. Zoom W into a W_* with two disks of
disjoint W_*-range. Then W_i and W_j have a transform-rigidity
conflict on those disks, contradiction.
So the exponential case cannot occur. The subexponential case is
harder; it takes PART VII's probability tree.
================================================================================
PART VII. THE SUBEXPONENTIAL CASE: A PROBABILITY TREE AND PAIR COUNTING
================================================================================
Now assume limsup_{j->infty} M(j)^(1/j) <= 1. The records do not
grow exponentially, so we cannot rely on local Holder continuity of u.
We must work directly with u, mu, and the level continua C_t.
The idea (from Cannon-Floyd-Parry and Bonk-Kleiner): distribute
"variation" of u across a TREE of small balls. The measure inequality
(MEAS-OSC) controls the interval lengths carried by disjoint balls.
The tree's transition probabilities have two quadratic bounds:
strict (at "good nodes" with two separated variation regions) and
weak (at "bad nodes"). Multiplied over a branch, they force a
definite fraction of probability onto good nodes at every depth.
At any common depth, we then compare many expansions that share
the SAME pattern of two target regions but have DISTINCT punctures.
This yields a measure contradiction (PART VIII).
Throughout: choose x_0 in U where u is NOT locally constant (i.e.,
x_0 in N(u)). This is possible because u is nonconstant and
continuous. Choose epsilon_0 > 0 with closure(B(x_0, 3 epsilon_0))
subset U. Set
K = Bbar(x_0, epsilon_0), (K)
K+ = Bbar(x_0, 2 epsilon_0), (K+)
and shrink epsilon_0 if necessary to choose z_0 in Z a positive
distance from K+. All tree centers lie in K; all enlarged source
neighborhoods lie in K+. Use the constants of MEAS-OSC for this
fixed region.
1. NODES, CHILDREN, AND GOODNESS
--------------------------------
A NODE is a triple (x, R, J) where:
- x in K is a center,
- R = r_k is a scale small enough that Bbar(x, c_0 R) subset K+
and (B_0 + 2) R < d_1/2 (so level continua exit neighborhoods),
- J subset (0,1) is a nonempty open interval with J contained in
(min_{Bbar(x, c_0 R)} u, max_{Bbar(x, c_0 R)} u). (STRADDLE)
Define Delta = |J| (length of the value interval).
Key consequence: for every t in J, a path in B(x, L c_0 R) STRICTLY
STRADDLES t (endpoints below and above t), so by (LEVEL-U) meets C_t.
Since diam(C_t) >= d_1, and (B_0 + 2) R < d_1/2, every C_t for
t in J must LEAVE Bbar(x, B_0 R) and MEET every intervening radial
sphere { d(x, \cdot) = b*R } for b in (a, B_0).
CANDIDATE SET: take Y = a maximal r-separated subset of the annulus
A_x(R) = { y : (A-1) R <= d(x,y) <= (B_0 + 1) R }. (ANNULUS)
This is an r-net in the annulus. Packing: #Y <= C_Y a^(Q D), where
D >= 1 is the DEPTH ratio (r = theta R with theta = a^{-D}).
For each y in Y, define range intervals
I_y^in = closed interval [ min u, max u ] on Bbar(y, 2r),
I_y^out = closed interval [ min u, max u ] on Bbar(y, c_0 r),
and put
J_y = J intersect int(I_y^out),
t_y = |J_y| / Delta. (RATIO)
So t_y in [0,1] measures "what fraction of J survives near y".
A positive t_y gives a nested child interval of J satisfying STRADDLE
for the child node (child center y, child scale r).
CHILDREN: nodes of form (y, r, J_y) with y in Y and t_y > 0. By
construction their intervals are nonempty and open, satisfy STRADDLE,
and J_y subset J.
GOODNESS: a node is GOOD if there are two candidates y^-, y^+ with:
|I_{y^-}^in intersect (j_-, j_- + Delta/4)| >= kappa Delta, (GOOD-IN-)
|I_{y^+}^in intersect (j_- + 3Delta/4, j_+)| >= kappa Delta, (GOOD-IN+)
dist(I_{y^-}^out, I_{y^+}^out) >= Delta/4, (GOOD-OUT)
where kappa = c_* a^{-Q D}, with c_* a small positive constant
(0 < c_* <= 1/(8 max{C_Y, 1})).
Read: GOOD means the child interval J has two candidate VARIATIONS
in its lower quarter and upper quarter that survive contraction
(inner intervals), and whose 2r-neighborhoods (outer intervals)
stay SEPARATED in value.
2. THE PROBABILITY FLOW: SETUP
-------------------------------
We build a probability distribution on the children of every
nonreached node, satisfying two-stage control.
Fix EXPONENTS
b_* = 1/2, (EXP-b*)
tau = 1/20, (EXP-tau)
beta = tau / (4 (Q+1)), (EXP-beta)
gamma = beta / 2. (EXP-gamma)
Let BAD nodes (not good) require transition probabilities p_y with
p_y <= a^{-gamma D} t_y, (BAD-LIN)
p_y <= a^{-b_* D} t_y^2. (BAD-QUAD)
Let GOOD nodes require
p_y <= a^{-gamma D} t_y, (GOOD-LIN)
p_y <= a^{tau D} t_y^2. (GOOD-QUAD)
Linear bounds multiply along a branch to make the branch probability
small. Quadratic bounds multiply along a branch to compare the
probability with the squared interval length.
3. GEOMETRIC CONSTANTS
----------------------
We fix the geometry (independently of D, which is chosen later).
Put c_in = 2L (inner path enlargement factor). Choose c_0 > c_in + 2
so large that, with
A = 4 L c_0 + 20, (GEO-A)
B_0 = A + 10, (GEO-B0)
c_pat = 2 C_E c_in / (A - 2)^2, (GEO-cpat)
we have
c_E c_0 / (B_0 + 2)^2 > 12 c_pat. (GEO-CONST)
This ratio grows linearly in c_0, so the choice is possible.
The constant c_pat is the target radius factor: paths joining extrema
of the two inner child balls will expand into balls of radius
c_pat theta/2, while the larger source balls will contain the
preimages of radius-10 c_pat theta target balls.
Next choose
H > max { 4(B_0 + 1), 4 C_mu c_0, 2 }. (GEO-H)
First inequality preserves separation BETWEEN branches; second makes
the measure enlargements at one generation disjoint.
c_* is fixed as specified above. Require theta < 1/4 and c_0 theta < 1.
All further LOWER BOUNDS on D depend only on this fixed data. The
uniform upper bound on node scales can be chosen now (before D)
because source radius factors are fixed: source neighborhoods and
measure enlargements stay in K+ for every node. The ROOT SCALE is
chosen after D has been fixed.
4. NONGOD NODES HAVE MANY LARGE CHILDREN
----------------------------------------
LEMMA (MANY CHILDREN AT A BAD NODE). For sufficiently large D, every
node that is NOT good has at least c_H a^D mutually H r-separated
candidates with t_y >= 1/8, where c_H > 0 is independent of the
node and D.
PROOF. Fix b in [A R, B_0 R]. Each C_t for t in J meets the sphere
d(x, \cdot) = b. So the inner intervals I_y^in of candidates with
| d(x,y) - b | <= r COVER J (as t_y varies over these candidates,
the range intervals cover the value interval J of the node).
By #Y <= C_Y a^{Q D}, two of them meet the lower and upper quarters
of GOOD-IN- and GOOD-IN+ with margin at least kappa Delta.
Suppose both t_{y^-}, t_{y^+} < 1/8. Then the outer interval
I_{y^-}^out ends below j_- + 3 Delta/8 (its intersection with J has
length < Delta/8 but meets the lower quarter; so the total intersection
length stays small). Similarly I_{y^+}^out starts above j_- + 5 Delta/8.
Their distance exceeds Delta/4, making the node GOOD by GOOD-OUT.
Contradiction with node NOT good. So one candidate has t_y >= 1/8.
Apply this at radial values spaced by (H + 3) r across [A R, B_0 R].
There are at least c_H a^D such values. The resulting centers (each
with t_y >= 1/8) are more than H r apart by construction. []
BAD NODE PROBABILITY RULE: at a bad node, put UNIFORM probabilities
on the candidates of MANY CHILDREN AT A BAD NODE. For large D they
satisfy BAD-LIN and BAD-QUAD:
p_y <= c_H^{-1} a^{-D} (since p_y = 1/(number of candidates)),
t_y >= 1/8,
p_y = O(a^{-D}) which is smaller than both a^{-gamma D} (1/8)
and a^{-b_* D} (1/64) for large D.
5. SEPARATED CAPACITY FOR GOOD NODES
-------------------------------------
For a good node we need a stronger mixture. This is the most
technical lemma of the whole proof.
LEMMA (SEPARATED CAPACITY). For sufficiently large D, uniformly over
all nodes, there is an H r-separated subset Y' subset Y such that
sum_{y in Y'} min{ a^{-gamma D} t_y, a^{tau D} t_y^2 } >= 1. (CAPACITY)
PROOF. Form a graph on Y by joining two candidates when their
distance is at most H r. Packing bounds its degree; so graph
coloring partitions Y into at most C_H separated classes, with C_H
independent of D.
Suppose the conclusion fails. Then
sum_{y in Y} min{ a^{-gamma D} t_y, a^{tau D} t_y^2 } < C_H. (FAIL-CAP)
Call y LARGE if t_y >= a^{-(tau + gamma) D} and SMALL otherwise.
On large candidates the minimum is the LINEAR one, so
sum_{large y} t_y <= C_H a^{gamma D}. (LARGE-LEN)
We will test levels at an intermediate radius s, larger than r but
smaller than R. This tests at more radial spheres than the large
ranges can occupy, and lets us detect the remaining levels with a
modulus cost at scale r/s.
REMOVE THE LARGE RANGES. Set D' = ceil(beta D) and s = a^{-D'} R.
For large D, 0 < D' < D and s < R/4. Choose s-spaced radial bins
B_subset in [ (A+1) R, (B_0 - 1) R ], so #B_subset is comparable to
a^{D'}. On B_subset x J use counting measure times Lebesgue measure.
Mark (b, t) if a LARGE candidate satisfies:
| d(x, y) - b | <= 5 s,
t in I_y^out.
Each large candidate meets at most 11 bins (5s spacing). By LARGE-LEN
and CLOSED-RANGE-LENGTH (the measure |J intersect I_y^out| = t_y Delta),
the marked measure is at most C C_H a^{gamma D} Delta.
Since gamma < beta:
unmarked measure >= c a^{D'} Delta. (UNMARKED)
DETECT THE REMAINING LEVELS. Take a maximal s-separated set Z_s in
A_x(R). Its radius-s balls cover the annulus; #Z_s <= C a^{Q D'}.
Expand at each z in Z_s at depth k + D' giving h_z. Put
q = r/s = a^{-(D - D')} and take macroscopic admissible weights
omega_xi, xi in T_{D-D'}, of squared mass M(D-D').
For fixed C_T > 2 + C_E, define
sigma_y^z = sum_{xi : d(xi, h_z y) <= C_T q} omega_xi,
if d(y,z) <= 3 s, else 0;
sigma_y = sum_{z in Z_s} sigma_y^z.
Within Bbar(z, 3s), the candidate images are separated by a fixed
multiple of q. Thus each sum has boundedly many terms; each target
weight is charged boundedly many times. Hence for each z,
sum_y (sigma_y^z)^2 <= C M(D-D'). (SINGLE-Z)
Each y lies within 3s of boundedly many auxiliary centers. Packing
and Cauchy-Schwarz:
sum_y sigma_y^2 <= C a^{Q D'} M(D - D'). (SIGMA-ENERGY)
Fix an UNMARKED (b, t). Choose w in C_t on the sphere d(x, \cdot) = b
and z in Z_s with d(w,z) <= s. The diameter bound gives a point of C_t
outside Bbar(w, 2s). Join these points in the r/4-neighborhood of C_t
(by the paths tool), stopping at the first radius-s hit about w.
The resulting path pi has diameter >= s and lies in
Bbar(w, s) subset Bbar(z, 2s). So the scale-s expansion h_z makes pi
MACROSCOPIC by the choice of d_0.
The radial bin lies between (A+1) R and (B_0 - 1) R, while s < R/4.
So every point of pi lies between radii (A+3/4)R and (B_0-3/4)R
about x. In particular, pi lies in the candidate annulus and has a
candidate center within distance r. The same fixed source margin
places every level point within r/4 of pi in K+ subset U. On these
points, the scalar-limit property gives u = t (even though it does not
assert this identity on parts of C_t outside U).
For a target cell meeting h_z pi, choose a preimage hit v in pi and
y in Y with d(y,v) <= r. Then d(y,z) < 3 s. If xi is the target
cell's center, the expansion formula gives d(xi, h_z y) < (2 + C_E) q
< C_T q, so its weight occurs in sigma_y^z.
A point of C_t is within r/4 of v, hence within 5r/4 < c_0 r of y.
The preceding interior check gives t in I_y^out. Since
| d(x,y) - b | <= s + r < 5 s, unmarkedness makes y SMALL.
Admissibility of omega therefore yields
sum_{small y : |d(x,y) - b| <= 5s} sigma_y * 1_{I_y^out}(t) >= 1
on every unmarked pair (b,t). (UNMARK-IND)
COMPARE THE ENERGIES. Integrating UNMARK-IND over B_subset x J
(counting measure times Lebesgue), using bounded bin multiplicity:
c a^{D'} Delta <= C Delta * sum_{small y} sigma_y t_y.
Only finitely many closed-interval indicators are integrated; no
measurable selection of continua or paths is needed. Singleton ranges
contribute zero. Cancel Delta > 0. Cauchy-Schwarz and SIGMA-ENERGY:
sum_{small y} t_y^2 >= c a^{-(Q-2) D'} / M(D - D'). (SMALL-SQ)
Now use SUBeXPONENTIAL growth: for every fixed tau > 0, we have
M(j) <= C a^{tau j / 8} for large j (by the definition of subexponential
growth). Then absorb C by increasing D, giving M(D-D') <= a^{tau D/4}
for large D.
Since D' <= beta D + 1 and (Q-2) beta < tau/4 (by choice of beta),
we get:
sum_{small y} a^{tau D} t_y^2
>= c a^{[3 tau/4 - (Q-2) beta] D}
>= c a^{tau D/2}. (SMALL-SQ-REDUX)
On small candidates, a^{tau D} t_y^2 is exactly the quadratic part of
the capacity. This contradicts FAIL-CAP. Every threshold for D is
uniform over nodes. []
FIXED D: At each good node, divide the positive capacities on a set
Y' from SEPARATED CAPACITY by their sum. These probabilities satisfy
GOOD-LIN and GOOD-QUAD (multiply both sides of the capacity by a
normalization, and note the min{} inequality translates to the pair
of bounds via t_y properties). At the remaining nodes, use the
uniform rule from MANY CHILDREN AT A BAD NODE. Every positive
transition retains a nonempty open child interval J_y subset J.
6. COMMON TARGET PATTERNS
-------------------------
The output of goodness is now padded into a finite-list target
pattern (independently of the node).
Fix a node's expansion h = v^{-1} when constructing it. At a good
node, join the extrema of each inner ball by a path in
B(y^±, c_in r). Each path has endpoint variation >= kappa Delta.
Its points lie at distance at least (A-2) R from x, so the expansion
formula puts its image within c_pat theta/2 of h y^±.
Conversely, write D_w = max{ R, d(x,w) }. If d(y,z) >= c_0 r, then
D_z <= D_y + d(y,z) and monotonicity of s / (D_y + s) give
d(h y, h z) >= c_E R c_0 r / ( D_y (D_y + c_0 r) )
>= c_E c_0 theta / (B_0 + 2)^2
> 12 c_pat theta. (EXIT-SEP)
Thus
Bbar( h y, 10 c_pat theta ) subset h( B(y, c_0 r) ). (PATTERN-INCL)
Choose a fixed finite c_pat theta-net in Z. For each h y^± choose a
net point within c_pat theta and take the closed radius-3 c_pat theta
ball about it. The ordered pair (B^-, B^+) is the node's PATTERN.
Each ball:
- contains its path image,
- lies in the corresponding outer source-ball image (PATTERN-INCL).
Consequently, for u o h^{-1}, the two whole-ball range intervals
have DISTANCE at least Delta/4, and each ball contains a path with
endpoint variation at least kappa Delta.
There are FINITELY MANY patterns, independently of depth (because the
net is fixed and c_pat theta is fixed once D is fixed).
7. THE TREE AND ITS FLOW
------------------------
Choose k_0 so large that R_0 = r_{k_0} satisfies all the source-scale
conditions above and
2 (B_0 + 1) R_0 < epsilon_0/2,
(B_0 + 2) R_0 < d_1/2. (ROOT-SCALE)
Non-local-constancy of u at x_0 supplies a nonempty open interval
J_0 subset (0,1) strictly inside the range on Bbar(x_0, c_0 R_0).
Write Delta_0 = |J_0| and construct the tree recursively, storing
each node's expansion and, at good nodes, its pattern.
Property 1: bounded displacement. A child moves by at most
(B_0 + 1) R from a parent of scale R. Consequently all descendants
stay within
(B_0 + 1) / (1 - theta) * R <= 2 (B_0 + 1) R (DESC-DISP)
of that parent, and all centers lie in K.
Property 2: separation. Generation n has scale r_{k_0 + D n} and its
centers are H times that scale apart. This holds for siblings by
construction. Children of distinct parents of scale R are separated
by at least
[ H - 2 (B_0 + 1) ] R > H R / 2 > H theta R. (CROSS-SEP)
Property 3: level energy. At any level of scale R, MEAS-OSC bounds
Delta_i^2 by C mu( Bbar(x_i, C_mu c_0 R) ). These enlarged balls are
disjoint because H > 4 C_mu c_0 (by choice of H). Hence
sum_i Delta_i^2 <= E_0, E_0 = C mu(Z) < infinity. (LEVEL-E)
Property 4: flow probabilities. Give the root probability 1 and let
P_i be the product of transitions to node i. Every level has total
probability 1. If i has generation n, let m_i count its ancestors at
generations 0, ..., n-1 that are NOT good. Multiplying the transition
bounds and telescoping the interval ratios yields
P_i <= a^{-gamma D n} Delta_i / Delta_0, (FLOW-LIN)
Delta_i^2 >= Delta_0^2 P_i a^{D((b_* + tau) m_i - tau n)}. (FLOW-QUAD)
Conditionally, if i is m generations below j, nesting gives
P_i / P_j <= a^{-gamma D m} Delta_i / Delta_j
<= a^{-gamma D m}. (COND-FLOW)
8. LEVELS WITH SUBSTANTIAL GOOD MASS
------------------------------------
LEMMA (LEVELS WITH SUBSTANTIAL GOOD MASS). Infinitely many
generations have total probability at least 1/2 on good nodes.
PROOF. By FLOW-QUAD, LEVEL-E, and Jensen's inequality:
E_0 / Delta_0^2
>= sum_i P_i a^{D((b_* + tau) m_i - tau n)}
>= a^{D((b_* + tau) sum_i P_i m_i - tau n)}.
Take logs (base a):
log_a (E_0 / Delta_0^2) >= D ( (b_* + tau) sum_i P_i m_i - tau n ).
Solve for the average sum_i P_i m_i:
(1/n) sum_i P_i m_i <= tau / (b_* + tau) + O(1/n)
= 1/11 + O(1/n).
(AVG-BAD)
(Because b_* = 1/2 and tau = 1/20, b_* + tau = 11/20, and
tau / (b_* + tau) = (1/20) / (11/20) = 1/11.)
Conservation of probability identifies (1/n) sum_i P_i m_i with the
AVERAGE BAD MASS over the first n generations (weighted by
probability).
If good mass were eventually less than 1/2 (so bad mass at least 1/2
at large n), that average (over generations) would have lower limit
at least 1/2. Contradiction with (AVG-BAD) which is 1/11. []
9. PROPERTIES OF THE FLOW
-------------------------
PROPOSITION (PROBABILITY FLOW). The construction above gives:
- a fixed D;
- a tree with generation-n centers that are H r_{k_0 + D n}-separated;
- descendant displacement DESC-DISP;
- nested nonempty value intervals;
- probabilities satisfying FLOW-LIN, FLOW-QUAD, COND-FLOW;
- infinitely many levels having GOOD MASS at least 1/2;
- every node having a fixed expansion;
- every good node having a pattern from a fixed finite list of
ordered target-ball pairs;
- for that expansion, each target ball containing a path with
endpoint variation at least kappa Delta_i;
- the two whole-ball range intervals having distance at least
Delta_i / 4;
- the target balls lying in the images of the respective outer
balls of radius c_0 theta r_{k_0 + D n}.
All the assertions were proven above. []
10. WHAT IS READY FOR PART VIII
-------------------------------
We now have:
- a probability tree with growing generations (n -> infinity),
- a set of GOOD nodes at various generations with mass >= 1/2,
- a fixed finite LIST of target patterns. At a good node, the
pattern (B^-, B^+) is an ordered pair of BALLS with the properties
in PROBABILITY FLOW.
PART VIII fixes a large generation n, takes a good-probability class
of nodes (dyadic in probability, and by pattern), and compares pairs
of nodes with DISTINCT punctures.
================================================================================
PART VIII. PAIR COMPARISON AND THE END OF THE SUBEXPONENTIAL CASE
================================================================================
We compare expansions at many nodes of a SINGLE generation n. The
statement to be contradicted:
PROPOSITION (NO SUBEXPONENTIAL RECORDS). The moduli M(n) cannot be
unbounded while satisfying limsup M(n)^(1/n) <= 1.
Setup: fix the flow of PART VII, including D, and a compact
neighborhood of its centers in U. Use the point z_0 chosen in the
flow construction, positive distance from that neighborhood. At
generation n write
k = k_0 + D n, R = a^{-k}, p_i = h_i z_0, (GEN-K)
where h_i = v_i^{-1} is the expansion already chosen for node i.
The centers x_i are H R-separated. From FLOW-LIN and COND-FLOW:
P_i <= a^{-gamma D n} Delta_i / Delta_0, (PAIRS-FLOW-1)
P_j / P_A <= a^{-gamma D m} if j is m generations below A. (PAIRS-FLOW-2)
We use the arbitrarily large generations having good mass at least
1/2 (from LEVELS WITH SUBSTANTIAL GOOD MASS).
1. A PROBABILITY CLASS AND SEPARATED PUNCTURES
-----------------------------------------------
Fix a generation n large enough that the conclusion of LEVELS WITH
SUBSTANTIAL GOOD MASS applies: the good nodes at generation n have
total mass >= 1/2.
There are at most C a^{Qk} nodes. Discard those with P_i < a^{-2Qk}
(loss at most C a^{-Qk} of total mass; so remaining good nodes have
mass at least 1/3 for large n). Partition the remaining nodes:
- by target pattern: finite many patterns;
- by DYADIC probability class: a node's probability P_i is dyadic if
P_i in [2^{-(m+1)}, 2^{-m}) for some integer m. Since
P_i >= a^{-2Qk}, only m in [0, 2Qk / log_2(a)] matter, i.e. O(k)
classes.
Pigeonhole: there is a single dyadic probability class I (also by
pattern) such that:
P_* <= P_i <= 2 P_* for all i in I, (PAIRS-CLASS)
sum_{i in I} P_i >= c / k. (PAIRS-CLASS-MASS)
Writing N = |I| (the number of nodes in the class), we have
N * P_* >= c / (2 k). (PAIRS-CLASS-SIZE)
Also P_i <= a^{-gamma D n} by PAIRS-FLOW-1, so
N >= c k^{-1} a^{gamma D n} -> infinity. (PAIRS-CLASS-GROW)
We will use punctures separated at scale a^{-l} much larger than R.
The final source-overlap estimate carries a factor a^{Ql}, so
arrange Ql <= gamma D n. Put
zeta = gamma / Q, m = floor(zeta n), l = D m. (PAIRS-L)
Both l and k - l tend to infinity.
2. BOUNDED PUNCTURE MULTIPLICITY
--------------------------------
LEMMA (PUNCTURE MULTIPLICITY). For every fixed C > 0, the balls
{Bbar(p_i, C R) : i at generation n} have uniformly bounded
multiplicity.
PROOF. Centers assigned to the same depth-k vertex have mutual
distance O(R). Their H R-separation and packing bound their number.
If several puncture balls share a point, their punctures have mutual
Gromov products at least k - O_C(1) (from the visual metric
comparing d(p_i,p_j) to a^{-k} scale). By the inverse prefix bound
(E6) and the prefixes lemma, all corresponding inverse vertices
v_i lie in one bounded Cayley-graph ball. There are boundedly many
such vertices, each with the bounded multiplicity just proved. []
3. PROBABILITY NEAR A PUNCTURE
------------------------------
LEMMA (NEAR-PUNCTURE PROBABILITY). For every node i at generation n:
sum_{j at gen n, d(p_j, p_i) <= a^{-l}} P_j
<= C a^{-gamma D m}. (NEAR-PUNCT-PROB)
PROOF. Fix an ancestor A at generation n - m; its depth is
k - l = k_0 + D (n - m).
CLAIM: only boundedly many descendants j of A satisfy
d(p_j, p_i) <= a^{-l}.
Split a geodesic word for v_j^{-1}:
v_j^{-1} = w_j u'_j, |w_j| = l, |u'_j| = k - l. (WORD-SPLIT)
The puncture condition (d(p_j, p_i) <= a^{-l}) and the inverse prefix
bound (E6) give:
(p_j | p_i)_e >= l - C,
(v_j^{-1} | p_j)_e >= k - C.
Thus w_j lies a bounded distance from the depth-l point on a fixed ray
to p_i. This leaves boundedly many possibilities for w_j.
Reverse and invert THE SAME word. This gives the geodesic
v_j = (u'_j)^{-1} w_j^{-1}. (WORD-REV)
Its depth-(k-l) prefix is therefore (u'_j)^{-1}. Since
d(x_j, x_A) <= 2 (B_0 + 1) r_{k-l}, (DESC-DIST)
thinness and the prefixes lemma put this prefix a bounded distance
from the depth-(k-l) point on a fixed ray to x_A. So u'_j also has
boundedly many possibilities.
Products of a bounded-distance vertex w_j and a bounded-distance
vertex u'_j determine boundedly many vertices v_j. The first step of
PUNCTURE MULTIPLICITY bounds the nodes assigned to each. We count
group elements here, so nonuniqueness of geodesic spellings causes
no additional multiplicity. Source and inverse prefixes TOGETHER
account for the full word.
Each such descendant j has P_j <= P_A a^{-gamma D m} by
PAIRS-FLOW-2. Sum first over these (boundedly many) descendants, then
over ancestors A at generation n - m, whose probabilities sum to one:
sum_{j near p_i} P_j <= sum_A P_A (C a^{-gamma D m})
= C a^{-gamma D m}. []
4. FAR PAIRS
------------
For each i in I, the fraction of the class with puncture distance at
most a^{-l} is bounded by
C a^{-gamma D m} / (N P_*)
<= C k a^{-gamma Dm} = o(1). (NEAR-FRAC)
This includes the diagonal j = i. So at least c N^2 unordered pairs
i != j satisfy
d_ij := d(p_i, p_j) > a^{-l}. (FAR)
Call them FAR PAIRS. Uniformly for these,
R / d_ij <= a^{-(k-l)} -> 0. (FAR-SCALE)
5. COMPARISON ON A COMPACT COMMON DOMAIN
----------------------------------------
Let K subset U contain all centers, with its closed epsilon-neighborhood
still in U. The upper puncture bound (E4) gives
d(h_i y, p_i) >= A_h R/2 ==> d(x_i, y) <= D_y <= 2C/A_h. (EXT-PRE)
Choose A_h so 2C/A_h < epsilon/2 and A_h/2 > 2C/d_1, and set
E_i = Bbar(p_i, A_h R).
Every h_i C_{N,t} reaches B(p_i, A_h R/2): the diameter bound supplies
a source point at distance >= d_1/2 from x_i, whose image has puncture
distance at most 2C R / d_1 <= A_h R/2. (DIAM-HIT)
The target balls (from the flow's PATTERN) lie a uniform distance
delta_* > 0 from p_i: their preimages lie in the outer source balls,
where D_y <= (B_0 + 2) R, so the lower puncture bound (E5) applies
for large n. Because the pattern is SHARED, this holds for every i
in I.
For large n, uniformly for far pairs:
2 A_h R < a^{-l}, (DISJOINT-E)
A_h R < delta_* / 2. (AWAY-BALLS)
Thus E_i, E_j are disjoint, each lies compactly in the other's
transformed domain by EXT-PRE, and the pattern balls avoid both.
These thresholds are independent of N (the scalar approximation
index of PART V).
6. THE PAIR ALTERNATIVE
-----------------------
LEMMA (PAIR ALTERNATIVE). Every far pair at a sufficiently large
generation has an endpoint i satisfying
osc_{h_i^{-1}(E_j)} u >= (kappa/16) Delta_i. (PAIR-ALT)
PROOF. Write F_i = u o h_i^{-1} and F_j = u o h_j^{-1} on their open
domains.
Case 1: osc_{E_i} F_j >= Delta_j / 16. Then endpoint j qualifies
directly (with kappa <= 1). Done.
Case 2: osc_{E_i} F_j < Delta_j / 16. Then one pattern ball B^s
(lower or upper from the flow) has its F_j-range interval at
distance at least Delta_j / 16 from the range on E_i. Indeed, if
NEITHER did, the two pattern intervals would have distance less than
3 Delta_j / 16, contradicting the flow's whole-ball separation by
Delta_j / 4. (PATTERN-SEP)
Node i supplies a compact path eta subset B^s with endpoint
F_i-gap at least kappa Delta_i. Suppose for contradiction that
osc_{E_j} F_i < kappa Delta_i / 4. (F-I-SMALL)
Remove this latter range interval from the endpoint gap of F_i on
eta. The remaining set is a union of open intervals; pick a
closed subinterval I subset (0,1) with positive length, strictly
between the endpoint values along eta and a positive distance from
the F_i-range on E_j. Its length need not be uniform over pairs, but
only depends on the pair (i,j), which we now FIX before letting
N -> infinity. (FIX-PAIR)
Put F_{i,N} = H_N o h_i^{-1} and F_{j,N} = H_N o h_j^{-1}. The
fixed range margins and local uniform convergence (scalar limit)
imply for every t in I and all sufficiently large N:
h_i C_{N,t} hits eta and avoids E_j. (LEVEL-HITS)
For the first assertion, the endpoint values of the preimage path
under PART V (LEVEL-U) still strictly straddle all of I, so the level
continuum applies. For the second, the range on E_j remains disjoint
from I.
By STOP-CONTINUUM (PART V.11), h_i C_{N,t} has a subcontinuum K_{N,t}
containing a hit on eta, meeting the sphere d(z, p_i) = A_h R, and
avoiding B(p_i, A_h R). It remains in the level F_{i,N} = t and
avoids E_j. In particular:
K_{N,t} subset K_ij
:= Z \ ( B(p_i, A_h R) union B(p_j, A_h R) ) compactly inside U. (COMPACT-K)
Compact containment follows from EXT-PRE with half the radius. The
sphere hit belongs to E_i; no identification of that sphere with the
boundary of a metric ball is needed.
Uniform convergence on eta union E_i preserves the F_j range gap,
giving
osc_{K_{N,t}} F_{j,N} >= Delta_j / 32
uniformly for t in I. (OSC-LOWER)
Now the finite-cell CROSS-ENERGY (PART V.10) applies with f = F_{i,N},
w = F_{j,N}, and the level family {K_{N,t}}:
(Delta_j/32) |I| <=
sum_{q : B(q, 2r_N) intersects K_ij} osc(B(q,2r_N)) F_{i,N}
* osc(B(q,2r_N)) F_{j,N}.
(CROSS-LHS)
For large N these closed cells lie in the common domain. Both squared
oscillation sums TEND TO ZERO by SMALL-E (PART V.6, local oscillation)
combined with SMALL-E-applied-through-the-transform-g^(-1). Then by
Cauchy-Schwarz:
(Delta_j/32)^2 |I|^2 <= E^(2)(F_i; K_ij) * E^(2)(F_j; K_ij)
-> 0. (CS-ZERO)
Contradiction with the positive left side. Therefore
F-I-SMALL fails, hence
osc_{E_j} F_i >= kappa Delta_i / 4. (PAIR-ALT-ACTUAL)
The argument applies separately to every pair above the geometric
threshold; no convergence rate uniform over pairs is required. So
PAIR-ALT holds. []
7. COST BALLS AND TOTAL MULTIPLICITY
------------------------------------
Now we turn PAIR-ALT into a measure bound and control the overlap of
the charged neighborhoods.
Two distinct indices have distinct roles:
- with i FIXED, the target punctures p_j control the number of
partners j;
- at a source point, separation of the centers x_i controls the
possible indices i.
Both estimates use the fixed enlargement constant C_mu from the
measure estimate (MEAS-OSC).
For an ordered far pair put z_ij = h_i^{-1}(p_j). The relevant source
scales are:
puncture denominator D_{z_ij} ~ R / d_ij, (COST-DENOM)
radius of the charged neighborhood R^2 / d_ij^2. (COST-RADIUS)
The radius-divided-by-denominator is O(R/d_ij) -> 0. This separation
of scales lets the FIXED enlargement C_mu not break image control.
If y in h_i^{-1} E_j, then d(h_i y, p_i) >= d_ij - A_h R >= d_ij/2,
whence D_y <= C R / d_ij. These points approach x_i uniformly, so
the two-sided puncture bound applies and gives
D_{z_ij} is comparable to R / d_ij. (DENOM-OK)
The lower expansion bound and d(h_i y, p_j) <= A_h R then give
d(y, z_ij) <= C R^2 / d_ij^2. (Y-Z)
Fix a sufficiently large C_2, put s_ij = C_2 R^2 / d_ij^2, so
h_i^{-1}(E_j) subset Bbar(z_ij, s_ij). (COST-CONT)
Choose the enlargement constant C_mu from MEAS-OSC (uniform on a
compact neighborhood of the centers). Define
D_ij = Bbar(z_ij, C_mu * s_ij). (COST-BALL)
Their centers stay in the compact neighborhood (of the tree centers)
and their radii tend uniformly to zero.
By PAIR-ALT and MEAS-OSC (applied at the appropriate i):
mu(D_ij) >= c kappa^2 Delta_i^2. (COST-LOWER)
The enlarged balls still have SMALL IMAGES. The function D_y is
1-Lipschitz, and
C_mu s_ij / D_{z_ij} <= C R / d_ij <= C a^{-(k-l)} -> 0. (ENL-SMALL)
Thus D_y is comparable to D_{z_ij} on D_ij, and the upper expansion
bound yields
h_i(D_ij) subset Bbar(p_j, C_3 R). (COST-IMAGE)
For fixed i, PUNCTURE MULTIPLICITY (with the C_3 radius) bounds the
multiplicity of {D_ij}_j as j varies. Also
D_ij subset Bbar(x_i, C_4 R a^l), (COST-SOURCE)
because d(x_i, z_ij) <= C R / d_ij and the cost radius
C_mu s_ij = o(R/d_ij). The H R-separation of the x_i and the packing
lemma bound the multiplicity of these balls by C a^{Ql}.
8. CONCLUSION OF THE PAIR COMPARISON
------------------------------------
Assign each unordered far pair {i, j} to a qualifying endpoint
(i.e., one where PAIR-ALT holds; when both qualify pick arbitrarily).
Upper bound. At any point, COST-SOURCE allows only C a^{Ql}
"paying indices" i (from different families), and COST-IMAGE allows
boundedly many partners for each. Integrating this multiplicity
bound (i.e., summing mu(D_ij) over all pairs) gives:
sum_{far pairs assigned to i} mu(D_ij) <= C a^{Ql} mu(Z). (COST-UPPER)
This counts the enlarged balls directly and does not use any
invariance of mu. The assignment may be arbitrary when both endpoints
qualify: all lower bounds hold for either endpoint, and the
multiplicity estimate already counts every possible ordered
contribution.
Lower bound. Every paying endpoint satisfies, by FLOW-LIN:
Delta_i^2 >= Delta_0^2 a^{2 gamma D n} P_i^2
>= Delta_0^2 a^{2 gamma D n} P_*^2. (PAYING-DELTA)
There are at least c N^2 far pairs, so COST-LOWER and PAIRS-CLASS-SIZE
give total cost:
c N^2 kappa^2 Delta_0^2 a^{2 gamma D n} P_*^2
>= c' k^{-2} a^{2 gamma D n}. (COST-LOWER-TOTAL)
Here we used N P_* >= c/(2k).
Conclude. Since kappa, Delta_0 > 0 are fixed and
Ql = Q D floor(gamma n / Q) <= gamma D n, (Ql-BOUND)
comparison of COST-UPPER and COST-LOWER-TOTAL yields:
c' k^{-2} a^{2 gamma D n}
<= C a^{Ql} mu(Z) <= C a^{gamma D n} mu(Z).
(COMPARISON)
Divide both sides by a^{gamma D n}:
c' k^{-2} a^{gamma D n} <= C mu(Z).
But mu(Z) is a FIXED positive number, kappa and c' fixed, and
k = k_0 + D n is increasing linearly in n. So for large n:
c' k^{-2} a^{gamma D n} -> infinity,
contradiction. No subexponential unbounded records can occur. []
9. FINISHING THE PROOF OF THE MODULUS BOUND
-------------------------------------------
Combine the two cases:
PART VI excluded exponential unbounded records (lambda > 1).
PART VIII excluded subexponential unbounded records (limsup <= 1).
Together: M(N) cannot be unbounded. Therefore
sup_n M(n) < infinity. (MOD-BOUND)
This is the analytic input for PART IV.6 (BK-BK).
In PART IX we feed MOD-BOUND to BK-BK, then verify the exact setup
and consequence statements (Sullivan-Tukia, proper action, etc.),
completing Cannon's conjecture.
================================================================================
PART IX. FINISHING: MAIN THEOREM AND CLASSICAL CONSEQUENCES
================================================================================
Everything is now ready. MOD-BOUND (PART VIII.9) is the entire
analytic input for Cannon's conjecture. This part builds the main
theorem from it, then records the standard consequences in the
torsion-free case.
1. PROOF OF CANNON'S CONJECTURE
-------------------------------
THEOREM. Let G be a hyperbolic group with partial G homeomorphic to
S^2. Then G admits a homomorphism rho:G -> Isom(H^3) whose action is
proper and cocompact, with finite kernel.
PROOF. FOUR STEPS.
Step 1. Set up Z and M(n).
As in PART IV.5: fix a visual metric d on Z = partial G, choose
T_n a maximal r_n-separated net, and define M(n) = Mod_2(F_0, T_n)
for paths of diameter at least d_0.
Step 2. Bound M(n).
By PART VIII.9 (combining PART VI and PART VIII), sup_n M(n) < infty.
This is MOD-BOUND.
Step 3. Uniformize the boundary.
Apply PART IV.6 (BK-BK uniformization, Hypotheses A and B verified).
We get a quasi-Mobius homeomorphism
phi: (Z, d) -> (S^2, round).
Step 4. Build the action.
Conjugate G by phi to get mathcal_G = phi G phi^(-1) acting quasi-
Mobiusly on S^2 with a common distortion (uniform quasi-Mobius in
g). Apply PART IV.7 (Sullivan-Tukia: quasiconformal structure
invariance + MRMT + circumcenter construction). We get:
- a single quasiconformal structure q on S^2 invariant under mathcal_G,
- a quasiconformal F: S^2 -> S^2 with F^* q_round = q,
- the conjugated group F mathcal_G F^(-1) acting by Mobius or
anti-Mobius maps,
- the extension to a homomorphism rho: G -> Isom(H^3),
- properness and cocompactness on H^3 (from T(Z) analysis),
- finite kernel (isometries of H^3 fixing every S^2 point are trivial,
so the kernel is finite by torsion-freeness of hyperbolic group
boundary actions).
Done. []
2. TORSION-FREE CASE: CLOSED HYPERBOLIC 3-MANIFOLD
--------------------------------------------------
COROLLARY (torsion-free). If G is a torsion-free hyperbolic group
and partial G homeomorphic to S^2, then G is the fundamental group
of a closed hyperbolic 3-manifold (possibly nonorientable).
PROOF. Let rho be the action of PART IX.1. Point stabilizers under a
proper action are finite. Torsion-free means the only finite
subgroup is {1}, so all stabilizers are trivial: the action is FREE.
A free, proper, cocompact isometric action of G on H^3 gives
H^3 / G, which is a closed hyperbolic 3-manifold with fundamental
group G. The orientability fails in general because rho may involve
orientation-reversing isometries; the quotient is then nonorientable. []
3. VIRTUAL STRUCTURE
--------------------
COROLLARY (virtual structure). Let G be a torsion-free hyperbolic
group with partial G homeomorphic to S^2. Then:
(a) Some finite-index subgroup of G is isomorphic to pi_1(Sigma)
semidirect Z for a closed connected orientable surface Sigma.
(b) G has positive virtual first Betti number.
(c) G is virtually compact special.
(d) G is residually finite.
PROOF. Let M = H^3 / G be the closed hyperbolic 3-manifold from
PART IX.2, and let G = pi_1(M).
Let M^+ be the orientation double cover of M, using M if M is
orientable, and let G^+ = pi_1(M^+). Then G^+ is normal of index at
most two in G.
(a) AGOL'S VIRTUAL FIBERING (Agol, Theorem 9.2 of "The virtual Haken
conjecture"). Applied to the closed orientable hyperbolic
3-manifold M^+: gives a connected finite cover N of M^+
fibering over the circle with closed, connected, orientable
fiber Sigma.
The fibration sequence gives a group extension
1 -> pi_1(Sigma) -> pi_1(N) -> Z -> 1,
i.e., pi_1(N) = pi_1(Sigma) semidirect Z with the action of Z
via the monodromy.
(b) The surjection pi_1(N) -> Z gives b_1(pi_1(N)) >= 1. Since
pi_1(N) is a finite-index subgroup of G (N is finite cover of
M^+ which is finite cover of M), G has positive virtual first
Betti number.
(c) BERGERON-WISE CUBULATION (Bergeron-Wise, Theorem 5.3 of "A
boundary criterion for cubulation"). Applied to G^+ (acting on
its sphere boundary, with the quasiconvex codimension-one
subgroups provided by Agol's work):
G^+ acts properly cocompactly on a CAT(0) cube complex.
Because G^+ is torsion-free, the action is free (point
stabilizers finite, torsion-free means trivial).
AGOL'S VIRTUAL SPECIALNESS (Theorem 1.1 of "The virtual Haken
conjecture"). Applied to G^+-cube: gives a finite-index subgroup
of G^+ with compact special quotient.
Since G^+ is finite-index in G, the same finite-index subgroup
of G^+ is also finite-index in G. So G is VIRTUALLY COMPACT
SPECIAL.
(d) RESIDUAL FINITENESS. AGOL'S SEPARABILITY (Corollary 1.3 of
"The virtual Haken conjecture") applied to the trivial
subgroup of G^+ gives: G^+ is residually finite.
To push to G: the indices are finite. Take G^+ as a finite-index
subgroup of G. For any nontrivial g in G:
- If g is not in G^+, use the finite quotient G/G^+ to
separate.
- If g is in G^+ and g is nontrivial, residual finiteness of
G^+ gives a finite-index normal subgroup of G^+ separating
g. The finitely many G-conjugates of this subgroup intersect
to a finite-index subgroup of G that separates g (because
each of the finitely many conjugates separates some conjugate
of g, hence g by normality after intersection).
- If g lies in no finite intersection, apply the above to
all finitely many conjugates.
In all cases, G is residually finite. []
4. WHAT WAS PROVED
------------------
Summary of the four main logical steps:
STEP A. (PART IV.5, IV.6) Bounded combinatorial 2-modulus of
macroscopic paths implies quasi-Mobius uniformization of
the boundary. This is Kennedy-Bourdon-based.
STEP B. (PARTS V-VIII) Bounded combinatorial 2-modulus holds.
Proved by contradiction: assuming M unbounded gives either
exponential records (contradicted in PART VI) or
subexponential records (contradicted in PART VIII).
STEP C. (PART IV.7) Quasi-Mobius structure -> proper cocompact
isometric action on H^3 via Sullivan-Tukia + MRMT.
STEP D. (PART IX.1-3) Assembly into the main theorem, and standard
corollaries (torsion-free closed 3-manifold, virtual
fibering, virtual specialness, residual finiteness).
The proof is complete.
5. THE OBJECTS IN ORDER
-----------------------
For a reader trying to assemble the whole proof from scratch:
Setup: Gamma(G,S), Z = partial G, d visual metric, a > 1, r_k = a^{-k}.
Macroscopic modulus: T_n maximal r_n-separated; F_0 = paths of
diameter >= d_0; M(n) = Mod_2(F_0, T_n).
Goal: sup_n M(n) < infty.
If not:
- pick RECORDS N (lambda > 1 exponential, or limsup <= 1
subexponential);
- crossing duality gives weights rho of tiny squared sum;
- scalar construction gives u : U -> [0,1] nonconstant, mu finite,
MEAS-OSC, LEVEL-U, CROSS-ENERGY, STOP-CONTINUUM, SMALL-E;
- exponential case: blowup from Holder regularity, ambiguity via
transform comparison, second blowup contradiction;
- subexponential case: probability tree of separated balls with
quadratic transition bounds; forces good nodes at all depths;
pair-comparison charges mu more than its overlap allows.
Conclusion: sup_n M(n) < infty.
Uniformize: apply BK-BK (Bourdon-Kleiner Corollary 3.5).
Straighten: apply Sullivan-Tukia + MRMT.
Consequences: torsion-free case gives closed hyperbolic 3-manifold
(possibly nonorientable); virtual Agol + Bergeron-Wise gives virtual
fibering, virtual specialness, residual finiteness.
Done.
================================================================================
PART X. GLOSSARY OF SPECIALIZED TERMS
================================================================================
Alphabetical. Each entry: what the term means, where it is used in
this document, and one sentence of "how to picture it".
ADMISSIBLE WEIGHT. Nonnegative function w on the cell net T_n such
that every path in a given family F has total weight (over cells met)
at least 1. See PART IV.5. Picture: a "tax" on cells big enough to
capture any path you might want to catch.
ANNULAR ESTIMATE. Lemma bounding oscillation of H_N across a small
ball by the mu-mass of a surrounding annulus. See PART V.7. Picture:
"the value of H_N can change only slightly inside B(x,R) because the
significant weights are all in the annulus around it".
APPROXIMATELY SELF-SIMILAR. Metric sphere X where every small ball,
rescaled to unit radius, is L-bi-Lipschitz to an open subset of X.
See PART III.8. Picture: you can't tell the scale by looking.
BANACH-ALAOGLU. Functional analysis result: bounded sets of finite
measures on a compact metric space are weak-star compact. Used to
extract mu from mu_N. See PART V.8.
BK-BK. Bonk-Kleiner uniformization theorem as sharpened by Bourdon-
Kleiner Corollary 3.5. Bounded combinatorial 2-modulus (plus self-
similarity and arcwise connectedness) implies quasi-Mobius
homeomorphism to round S^2. See PART IV.6.
BLOWUP. Amplification of a Holder function by a scaling factor
sigma_k. See PART VI.1. Picture: zooming a photograph so a tiny
variation fills the frame.
CANDIDATE SET Y. An r-separated net in the annulus A_x(R). Used in
PART VII.1. Picture: all possible positions of a "child center" near
a parent.
CELLS. The closed balls B(q, 2r_n) for q in T_n. Cover Z with
bounded overlap. See PART IV.5.
CMU (C_MU). The enlargement constant in MEAS-OSC: (osc u on
B(x,s))^2 <= C_mu * mu(B(x, C_mu s)). Fixed once the scalar limit
is extracted. See PART V.8.
CONTINUUM. Nonempty compact connected set. NOT necessarily path-
connected. Z can be homeomorphic to S^2 and still have "fat"
continua that are not paths.
CROSS-ENERGY. Lemma: if a level family {D_t} of a continuous f has
bounded w-oscillation on every level, then the cell oscillation
energies of f and w multiply to at least b_0 |I|. See PART V.10.
This is the "sweeping integral" fact.
CROSSING DUALITY. Mod_2 of left-to-right crossings = 1 / ||P||^2
for a special least-norm point P in the convex hull of crossing
vectors; and P itself is admissible for the transverse (top-to-
bottom) family. See PART V.3.
CT (C_t). Continuum in Q with u = t on C_t intersect U, diameter
d_1, meeting every straddling path. See PART V.9. The "level set
as a barrier".
DIAGONAL APPROXIMANTS. Lemma building increasing N_j so that
f_j = (H_{N_j} o h_j^{-1} - u(x_j)) / sigma_j converges locally
uniformly to W with vanishing fixed-transform energies. See PART
VI.2.
DILATATION (of a quasiconformal map). K where |Df|^2 <= K det Df.
Ratio of singular values of Df is at most sqrt(K). See PART III.9.
EXPANSION (map h = v^{-1} at (x,k)). Magnification of the ball
B(x, r_k) to a Z-sized set. Behaves "bi-Lipschitz with scale R^{-1}"
on that ball. See PART IV.1.
EXPANSION FORMULA. (E1): c_E R d(y,z)/(D_y D_z) <= d(hy,hz) <=
C_E R d(y,z)/(D_y D_z) with D_y = max{R, d(x,y)}. The single most
used estimate in the paper.
F_0. Family of all paths in Z of diameter at least d_0. See PART
IV.5. The "macroscopic path family".
GROMOV PRODUCT. (y|z)_o = (d(o,y) + d(o,z) - d(y,z)) / 2. "How much
y and z overlap along rays from o". See PART III.4.
H_N. The regularized scalar approximant. H_N = sup over x-to-bottom
paths of inf along them of F_N, where F_N = inf over x-to-top paths
of max along them of U^0_N. See PART V.5. Picture: "two-layer flood".
HOLDER MAGNIFICATION. Lemma extracting a nonconstant blowup W of a
locally Holder function on a punctured sphere. See PART VI.1. Uses
eta > Q in the definition of sigma_k.
HYPERBOLIC GROUP. Finitely generated G whose Cayley graph has
uniformly thin geodesic triangles. See PART III.2.
KAPPA (kappa). The small constant c_* a^{-Q D} in the GOOD
condition. See PART VII.1. Tracks "how much of Delta survives at
a candidate".
LEVEL CONTINUA. Lemma: {H_N = t} contains a lateral continuum
C_{N,t} of diameter d_1. See PART V.5.
LOC-OSC. Local oscillation estimate: osc of H_N on B(x, Jr_N) <=
sum of rho-q for q near x. See PART V.6.
MEAS-OSC. Measure estimate: (osc u on B(x,s))^2 <= C mu(B(x, C_mu
s)). See PART V.8.
MOD-BOUND. sup_n M(n) < infinity. The analytic input for PART IV.6.
MOD_2. Combinatorial 2-modulus (defined in PART IV.5). Minimum
squared-sum over admissible weights.
MRMT. Measurable Riemann Mapping Theorem of Ahlfors-Bers. Solves
Beltrami equation for measurable coefficients. See PART III.9.
M(N). Mod_2(F_0, T_N) at record modulus index N. See PART IV.5.
N(u). Set of points where u is not locally constant. See PART VI.3.
NODE. (x, R, J): center x in K, scale R = r_k, open value interval
J subset (0,1) inside the range of u on Bbar(x, c_0 R). See PART
VII.1.
NETS. T_n is the maximal r_n-separated subset of Z. See PART IV.5.
PACKING. Lemma: #(s-separated set in B(x, t)) <= C (t/s)^Q. See
PART IV.3.
PATTERN. An ordered pair (B^-, B^+) of closed balls about net points
containing the variation paths of a good node after expansion. See
PART VII.6.
PROBABILITY FLOW. Transition probabilities on the tree of nodes,
satisfying linear and quadratic bounds. See PART VII.7.
PUNCTURE P = h z_0. Image of a fixed far point z_0 under the
expansion h. Stays away from h's center. See PART IV.1 and PART
VI.2.
QUASI-MOBIUS. Homeomorphism f of a metric space is quasi-Mobius if
the cross-ratio is distorted by a uniformly bounded factor. See
PART III.6.
QUASICONFORMAL. Homeomorphism f:Omega -> Omega' with f in W^1,2_loc
and |Df(x)|^2 <= K det Df(x) a.e. See PART III.9.
RECORDS N. Indices with M(N) -> infinity and M(N-l) <= lambda^{-l}
M(N) for 0 <= l <= N. See PART V.0.
REGULARIZATION. Passing from U^0_N to F_N and H_N via path envelopes
to get separated level sets. See PART V.5.
SELF-SIMILARITY. See APPROXIMATELY SELF-SIMILAR above.
SEPARATED CAPACITY. Lemma supplying enough radius-weighted capacity
at good nodes for the quadratic bound GOOD-QUAD. See PART VII.5.
SMALL-E. Estimate: fixed-transform energy of H_N tends to zero as
N -> infinity. See PART V.6.
STOP-CONTINUUM. Lemma: a continuum that starts inside a ball and
crosses its sphere has a subcontinuum that reaches the sphere while
avoiding the ball. See PART V.11.
STRADDLE (a node's interval J). J subset (min u, max u) on B(x,
c_0 R). Every t in J is traversed by a path crossing B(x, Lc_0 R).
See PART VII.1.
SULLIVAN-TUKIA. Method (Sullivan 1981 + Tukia 1980) for extracting
an invariant conformal structure from a quasi-Mobius group of S^2,
then applying MRMT to get a Mobius group. See PART IV.7.
SUPPORT (of mu). The support of mu lives on the closure of {q in
T_N for (some subsequence of) N}. See PART V.8.
TANGENT (to a quasiconformal map). The differential Df at a
differentiability point. Quasiconformality bounds its distortion at
a.e. points. See PART IV.7.
TREE. The probability tree of nodes with transitions P_i. See PART
VII.7.
TWIN RADII. R = r_l and s = r_{l+j} (or similar); the "nested
radius" structure of estimates. See PART V.7, VII.5.
U^0_N. Raw scalar: min{1, b_N(x)}. Interpolated continuously. See
PART V.4.
VISUAL METRIC. d(x,y) comparable to a^{-(x|y)_e}. See PART III.5.
WEAK CONVERGENCE. mu_N -> mu means integral f dmu_N -> integral f
dmu for every continuous bounded f. See PART V.8.
Z. Boundary at infinity of G. See PART III.3.
================================================================================
PART XI. CONCRETE CALCULATIONS FOR A READER TO DO BY HAND
================================================================================
These are worked examples on very small models, so a reader can
verify the definitions are doing what they should. Nothing here is
part of the proof; it is a way to build intuition and check
understanding. Do them with pencil and paper.
1. THE BOUNDARY OF Z, TWO POINTS
--------------------------------
Let G = Z (infinite cyclic) with S = {+1, -1}. Cayley graph is a
bi-infinite line. Its boundary is a two-point set: Z_G = {+infinity,
-infinity}.
Take the visual metric: d(+inf, -inf) = 1 (there is just one pair of
distinct points), and d(x, x) = 0. The combinatorial 2-modulus at
scale r_n:
T_n = r_n-separated maximal set of Z_G. Since Z_G = {+inf, -inf}
and r_n = a^{-n} -> 0, for large n both points are separated
(distance 1 > r_n). So T_n = {+inf, -inf}.
F_0 = paths in Z_G of diameter at least d_0 (with d_0 = 1/10, say).
A "path" in a two-point set is either the constant path at one
point (diameter 0) or a path from one point to the other (diameter
1). None of these is continuous in a nontrivial way (the two points
are isolated).
M(n) = Mod_2(F_0, T_n) = inf over admissible w: sum_{q in T_n}
w(q)^2. A path from +inf to -inf has diameter 1 >= d_0, so it
must "meet" cells at both points. Admissible means w(+inf) +
w(-inf) >= 1 (that is the trace on a diameter-1 path). Minimize
w(+inf)^2 + w(-inf)^2 subject to w(+inf)+w(-inf) >= 1: solution
at equality with both = 1/2, giving M(n) = 1/4.
So M(n) = 1/4 for all large n. Bounded, as the theorem requires.
Check the corollary: Z_G = two points, NOT homeomorphic to S^2, so
Cannon's conjecture does not apply here. The example just shows
that M(n) can be computed.
2. THE BOUNDARY OF F_2, A CANTOR SET
-------------------------------------
Let G = F_2 with S = {a, a^{-1}, b, b^{-1}}. Cayley graph is the
4-regular tree.
Rays from the identity are exactly the infinite reduced words in
{a, a^{-1}, b, b^{-1}} with no cancellation. Two rays are equivalent
iff they eventually agree (after no cancellations). So Z_G is the
set of infinite reduced words, which is homeomorphic to a Cantor
set.
With the visual metric, if two boundary points share their first m
letters but disagree at the (m+1)-st letter, then (x|y)_e = m, and
d(x,y) ~ a^{-m}.
For a specific number: take a = 2. So r_m = 2^{-m}. Take x = aaaa...
and y = aaab... (agree for 3 a's, then diverge). Then (x|y)_e = 3
and d(x,y) ~ 2^{-3} = 1/8.
PACKING: at depth k, a ray has 4 choices for its next vertex, but
one of them is the immediate "reverse" (back toward e), so ~3
forward choices. Between any two depth-k vertices, the map is
roughly uniform. Following PART IV.3, the packing exponent Q for
F_2's boundary is Q = log_a(A) where A = branching = 3, so
Q = log_2(3) ~ 1.58. This is the Hausdorff dimension of the Cantor
set under this metric.
Check the volume-growth bound: #(s-separated set in B(z, t)) ~
(t/s)^Q. With Q = 1.58, doubling is quite tame.
3. A TINY MODULUS PROBLEM BY HAND
---------------------------------
Take Z = [0,1] (an interval, NOT a sphere; we're just studying the
modulus). Let T_n = {0, 0.5, 1} (three cell centers). Cells are
[0, 0.5], [0, 1] -- well, "balls" B(0, 2r_n) for small r_n around
each center. Admissible weights for the family F_0 = macrosopic
paths (a path from one end to the other, say).
Say the path [0,1] passes through all three cells. So admissible
means w(0) + w(0.5) + w(1) >= 1. Minimize sum of squares: equality
with all three at 1/3, giving modulus 1/3.
So M(tiny) = 1/3 for this toy setup. That is essentially the
"effective modulus of one long path".
4. CROSS-RATIO INVARIANCE UNDER MOBIUS
--------------------------------------
For z in C-hat = C union {infinity}, a MOBIUS map is
z -> (a z + b) / (c z + d)
with ad - bc != 0.
A classical fact: the CROSS-RATIO
CR(x,y,z,w) = |x-z| |y-w| / (|x-w| |y-z|)
is INVARIANT under Mobius maps (up to orientation):
CR(f x, f y, f z, f w) = CR(x,y,z,w) for f Mobius.
WORKED EXAMPLE: f(z) = 1/z. Take x=1, y=2, z=3, w=4 in R.
CR(1, 2, 3, 4) = |1 - 3| |2 - 4| / (|1 - 4| |2 - 3|)
= 2 * 2 / (3 * 1) = 4/3.
Apply f: f(1)=1, f(2)=1/2, f(3)=1/3, f(4)=1/4. Compute directly:
CR(1, 1/2, 1/3, 1/4)
= |1 - 1/3| |1/2 - 1/4| / (|1 - 1/4| |1/2 - 1/3|)
= (2/3)(1/4) / ((3/4)(1/6))
= (1/6) / (1/8) = 8/6 = 4/3.
Yes, invariant. This is why CR is the natural quantity for "how
nonconformal is f": a conformal map preserves CR exactly; a
quasi-Mobius map distorts it by a bounded factor.
EXERCISE FOR THE READER: try f(z) = 2z + 1 with the same points.
Verify CR is invariant again. Then try f(z) = z-bar (complex
conjugate) and see CR(fx,fy,fz,fw) = CR(x,y,z,w), NOT its negative.
Reflections preserve cross-ratio up to modulus.
5. THE QUASICONFORMAL DILATATION ON A CONCRETE EXAMPLE
-----------------------------------------------------
Let f(z) = z + epsilon * zbar, for small epsilon in (0, 1). In real
coordinates (u = Re z, v = Im z):
Re f = u + epsilon u = (1+epsilon) u,
Im f = v - epsilon v = (1-epsilon) v.
So Df(x+iy) is the diagonal matrix diag(1+epsilon, 1-epsilon). Its
singular values are 1+epsilon and 1-epsilon. The ratio is
(1+epsilon)/(1-epsilon). Squared, that is the K-dilatation:
K = ((1+epsilon)/(1-epsilon))^2.
For epsilon = 0.1: K = (1.1/0.9)^2 ~ 1.49. For epsilon = 0.5:
K = 9. For epsilon close to 1, K -> infinity (f is close to
noninvertible).
If f is quasiconformal, then in the Beltrami equation f_z = mu(z)
f_zbar with |mu| < 1 a.e., we have |mu| = (K-1)/(K+1). For epsilon
= 0.1: |mu| = 0.49/2.49 ~ 0.20.
The MRMT says: pick any mu in L^infty with |mu| < 1 a.e., solve
f_z = mu f_zbar. There is a unique-up-to-Mobius quasiconformal
homeomorphism f. That is the "measure of ill-behavedness" that MRMT
can construct.
6. A SIMPLE PROBABILITY-FLOW COMPUTATION
----------------------------------------
Consider a very small toy version of PART VII: only TWO generations
of the tree, with 2 children at each node.
At the root (generation 0), the value interval J_0 = (0.4, 0.6) of
length Delta_0 = 0.2.
Children at generation 1:
- Child 1: value interval J_1 = (0.4, 0.5), ratio t_1 = 0.5,
probability p_1 = 0.5.
- Child 2: value interval J_2 = (0.5, 0.6), ratio t_2 = 0.5,
probability p_2 = 0.5.
Both good? Check: at child 1's level, "good" for the root would
need two separated variations. Here children 1 and 2 have DISJOINT
intervals (0.4,0.5) and (0.5,0.6), so the root is definitely
"not good" (no two children with a large outer overlap).
Actually we can just run through FLOW-LIN and FLOW-QUAD with
hypothetical exponents gamma D = 0.05, b_* D = 0.3, tau D = 0.02,
and the ROOT being "not good":
FLOW-LIN at root (bad): p_1 <= a^{-0.05} t_1 = 2^{-0.05} * 0.5
~ 0.966 * 0.5 = 0.48.
0.5 <= 0.48? NO. FLOW-LIN FAILS.
So this "equal-probability, half-split" choice is not
admissible for a bad node. We should have used uniform on more
than two children, or made the interval steeper.
The point: FLOW-LIN / FLOW-QUAD are STRINGENT. They don't allow
"lazy" splits. Each of the exponent choices must actively cost you
linear/quadratic mass when you pass to a child. That is what forces
good nodes to dominate at every level.
7. THE HOLDER MAGNIFICATION EXPONENT
------------------------------------
In PART VI.1, sigma_k = A_k / delta_k^eta with eta > Q. If Q ~
1.58 (F_2's boundary, PART XI.2), then a choice like eta = 2 is
permissible.
Then delta_k / r_k grows like r_k^{Q/eta - 1} = r_k^{1.58/2 - 1}
= r_k^{-0.21}, which goes to infinity. This makes
h_k^{-1}(K) subset Bbar(x_k, r_k) UNIFORMLY in k for compact K off
the puncture, which is what HOLDER MAGNIFICATION needs.
EXERCISE: check that if eta > Q, the exponent Q/eta - 1 < 0, hence
delta_k / r_k -> infinity. And if eta = Q exactly, delta_k / r_k is
bounded, and the argument fails.
8. THE RATIO 1/11 IN PART VII
-----------------------------
In PART VII.8, LEVELS WITH SUBSTANTIAL GOOD MASS, the bound
(1/n) sum_i P_i m_i -> 1/11
comes from tau / (b_* + tau) with b_* = 1/2 and tau = 1/20:
tau / (b_* + tau) = (1/20) / (1/2 + 1/20)
= (1/20) / (11/20)
= 1/11.
So on average, at most 1/11 of a node's "bad generations per step"
count against you. Since 1/11 << 1/2, at most 1/11 of the
probability can be bad in average. So at least 10/11 of the
probability is GOOD. In particular, at least 1/2 (by a huge margin)
is good. This is what LEVELS WITH SUBSTANTIAL GOOD MASS gives.
The choice b_* = 1/2 is arbitrary; any b_* > 0 works. The
important quantity is tau, which must be small, because tau controls
the quadratic slack at good nodes. Specifying tau = 1/20 ensures
the growth rate is sub-1, but any tau < 1 works.
9. WHAT A READER CAN NOW DO
---------------------------
After reading PART XI, the reader can:
- compute the combinatorial 2-modulus of a simple configuration;
- verify cross-ratio invariance under Mobius f(z) = 1/z;
- compute a quasiconformal dilatation for f(z) = z + epsilon zbar;
- check the growth of eta > Q in HOLDER MAGNIFICATION;
- trace the 1/11 ratio through PART VII.8.
If all of these are comfortable, the rest of the proof is "just"
tracking bookkeeping through deeper technical details. If any is
uncomfortable, re-read the relevant part of PART III.
END OF DOCUMENT
================================================================================
NOTES FOR THE READER
================================================================================
WHAT TO READ FIRST.
If you have never seen a Cayley graph, start with PART I.
If you want to see the shape of the proof immediately, read PART II.
If you already know the definitions, jump straight to PART IV.
THE PROOF HAS FOUR BIG MOVES.
(1) Reduce to a combinatorial 2-modulus bound (PART IV).
(2) Assume the bound fails (PART V).
(3) Kill the "exponential" alternative (PART VI).
(4) Kill the "subexponential" alternative (PART VII-VIII).
WHY SO MUCH MACHINERY.
The boundary is only KNOWN to be homeomorphic to S^2, not
necessarily round. So the proof has to work with a badly-behaved
metric. The combinatorial modulus is a way to measure "how badly
the boundary curves behave at small scales" without ever fixing a
particular metric or a particular round structure. The modulus
bound then says "the boundary is flexible enough to be rounded to
the standard sphere." From there, the classical Sullivan-Tukia
argument does the rounding.
WHAT A READER MUST ACCEPT ON FAITH.
- MRMT (Ahlfors-Bers): a Beltrami equation has a unique solution
class. This is a real theorem, proven in several textbooks.
- Sullivan-Tukia's construction of the invariant conformal
structure. This is a real theorem (both papers).
- Agol's virtual fibering theorem, Bergeron-Wise, and Agol's
residual finiteness.
- BK-BK's uniformization criterion and Bourdon-Kleiner's
Corollary 3.5.
- Basic quasiconformal machinery (Sobolev regularity,
a.e. differentiability, etc.), all classical.
WHAT A READER IS ASKED TO CHECK.
- The expansion formula (PART IV.1) with its three cases.
- Packing and paths (PART IV.3-4) from graph combinatorics.
- Crossing duality (PART V.3) as a convex optimization argument.
- The measurable Riemann mapping theorem's exact statement (PART
III.9).
- STABILIZER growth bound (PART VI.4) via base-change and the
loxodromic growth argument.
- The capacity lemma (PART VII.5), which is the most technical.
The rest is bookkeeping. It is long, but every step is under two
hundred words of argument.
================================================================================
END
================================================================================