================================================================
AN IRRATIONAL-TRACE COUNTEREXAMPLE TO THE REDUCED BAUM-CONNES
CONJECTURE
A complete, self-contained, plain-text proof.
================================================================
This file rewrites the original LaTeX paper in plain prose so that a
strong high-school reader who will look up a few terms can follow the
whole argument. Every lemma is stated and proved. Where we import a big
theorem we say exactly which one and why it applies.
The two ingredients that make the construction work are:
(1) TRACE TRICK. Luck's theorem says the canonical trace of any class
coming from the assembly map is rational. So ONE projection in a
reduced group C*-algebra with irrational trace has a K-theory
class outside the assembly image, and assembly is not onto. That
is how the conjecture is disproved.
(2) GEOMETRY PLUS LOCAL RULES. We build a finitely generated group
whose reduced C*-algebra contains such a projection. The group is
a "lamp" of a finite-field module over a Cayley graph built from
labelled expander graphs. Exact local module relations plus one
global geometric estimate give a uniform spectral gap. The gap
lets continuous functional calculus produce the projection INSIDE
the C*-algebra (not merely in a von Neumann algebra, which would
be too weak).
Organization, parts 0 through 13:
0 statement and shape of the argument
1 Luck's trace theorem (imported)
2 labelled expanders (imported, plus elementary checks)
3 Cayley graph, copies, coning
4 hyperbolicity of the coned graph (diagram argument)
5 avoidance and simultaneous overlap
6 codes over F_p
7 the module extension and its exact relations
8 the compact dual and uniform marginals
9 the operator and the enabled graph
10 deficits (local inside a copy, global by cycle rank)
11 spectral gap and the projection
12 the trace is irrational
13 conclusion: assembly fails
================================================================
PART 0. WHAT IS BEING PROVED, AND THE SHAPE OF THE ARGUMENT
================================================================
STANDING NOTATION. G is a countable discrete group. C*_r(G) is the
reduced group C*-algebra: the norm-closure of the complex group ring
C[G] acting on l^2(G) by the left regular representation. Its canonical
trace is
tau_G( sum_g a_g g ) = a_1, tau_G(1) = 1.
For a projection b in C*_r(G) (b = b* = b^2, or a finite matrix
projection), tau_G(b) is a real number in [0,1].
THE ASSEMBLY MAP. The reduced Baum-Connes assembly map is
mu_j : K_j^G( E~G ) -> K_j( C*_r(G) ), j = 0, 1,
where E~G ("underline E G") is the classifying space for proper
G-actions: fixed-point sets of finite subgroups are contractible and
those of infinite subgroups are empty. The domain is equivariant
K-homology with G-compact support, carried by a direct limit over
closed invariant G-compact subspaces of a proper G-CW model. Assembly
sends a geometric class to its analytic index. We use complex K-theory
with the trivial coefficient algebra C. No finite-dimensional or
cocompact model for E~G is assumed.
BAUM-CONNES CONJECTURE. mu_0 and mu_1 are isomorphisms.
MAIN THEOREM (proved in this file). There is a finitely generated
countable discrete group G_sur together with a genuine projection
b_0 in C*_r(G_sur), tau_{G_sur}(b_0) irrational,
whose degree-zero K-theory class satisfies
[b_0] is not in mu_0^{G_sur}( K_0^{G_sur}(E~G_sur) ).
Hence assembly is not surjective in degree zero and the
coefficient-free reduced Baum-Connes conjecture is FALSE.
THE ARGUMENT IN ONE PARAGRAPH. Build G_sur = V rtimes Gamma, where Gamma
is a finitely generated group whose Cayley graph contains translated
expanders as rigid "copies", and V is a quotient of the finite-support
F_p-module over Gamma by relations sitting on every translated copy. On
the compact dual Omega of V we get a local rule: an "enabled graph"
G_omega on Gamma. Copies carrying the prescribed label pattern become
isolated expander components and produce a kernel; everything else is
controlled by a uniform spectral gap. The gap is proved by a cycle-rank
argument charging the ranks of copy coordinates to pairwise disjoint
vertex sets. Because the gap is uniform in omega, the indicator of zero
is a continuous function of the operator, giving b_0 in C*_r(G_sur). Its
trace is the series sum_i p^{-k_i}, whose base-p digits are not
eventually periodic, hence irrational. Luck's theorem then finishes.
WHY COEFFICIENTS MATTER (context only). Positive results: Higson-
Kasparov give assembly WITH coefficients for groups acting properly by
affine isometries on Hilbert space (so all countable amenable groups);
Mineyev-Yu give the coefficient-free case for hyperbolic groups and
their subgroups; Lafforgue gives WITH coefficients for hyperbolic
groups. On the negative side, Higson-Lafforgue-Skandalis give
counterexamples with commutative COEFFICIENTS using expanders, but not
for the coefficient-free statement. Our counterexample is coefficient-
free. Purely von Neumann irrational-kernel constructions (Austin;
Pichot-Schick-Zuk; Grabowski) do not settle the C*-algebra question,
because their projections need not lie in C*_r(G). Getting the
projection inside C*_r(G_sur) is exactly what the uniform gap buys.
================================================================
PART 1. LUCK'S TRACE THEOREM: ASSEMBLY CLASSES HAVE RATIONAL TRACE
================================================================
The imported fact that turns an irrational trace into a counterexample.
No surjectivity of assembly is assumed.
LEMMA 1.1 (Luck; trace of assembly classes). For every countable
discrete group G,
tau_{G*}( mu_0^G( K_0^G(E~G) ) ) is contained in Q,
where tau_{G*} is the map on K_0 induced by tau_G (extended to matrices
as the sum of diagonal traces).
PROOF. Put Lambda_G = Z[ 1/|F| : F <= G finite ] inside Q. Luck's
theorem [Luck2002, Theorem 0.3] states that, after extending scalars to
Lambda_G, the trace image of assembly is all of Lambda_G. On an integral
class x the composite
K_0^G(E~G) --mu_0--> K_0(C*_r G) --tau_{G*}--> R
sends 1 (x) x to tau_{G*}(mu_0(x)), which therefore lies in Lambda_G,
a subgroup of Q. No surjectivity of assembly and no injectivity of
localization is used.
For completeness, our K-homology is the same as Luck's. Our domain is
the direct limit of KK_j^G(C_0(Z), C) over closed invariant G-compact
subspaces Z of a proper G-CW model, with assembly given by reduced
Kasparov descent times a cutoff projection, then this direct limit.
Luck uses the same proper equivariant KK-homology and analytic index.
His cocompact subcomplexes are cofinal for these supports: the quotient
is a CW complex and its compact subsets lie in finite subcomplexes.
Standard comparisons of the Baum-Connes and Davis-Luck formulations
[Hambleton-Pedersen 2004, Cor. 8.4; Kranz 2021, Thm. 5.3] complete the
identification. QED
Direct consequence. A class with irrational trace is not in the image of
mu_0. So it suffices to find ONE projection b_0 in C*_r(G) with
tau_G(b_0) irrational. Everything below builds such a b_0.
================================================================
PART 2. THE EXPANSION DATA: LABELLED EXPANDERS
================================================================
Constants, fixed in this order:
- d >= 3, the common degree.
- rho > 0, the common spectral gap of the expanders (below).
- eta = 1/100.
- kappa = rho/2, c = (eta/2) * min(kappa, 1).
- lambda > 0 with lambda < 1/24 and
12 (L_0 + 1) lambda <= 1/16, (2.1)
where L_0 is the universal constant defined later in (5.1). This
fixes lambda but NOT yet the alphabet or the arithmetic field.
PROPOSITION 2.1 (choice of data). There exist finite connected simple
d-regular graphs Theta_i (i = 1, 2, ...) of girth sigma_i with the
properties below. Write n_i = |Vert(Theta_i)|, k_i = floor(n_i/2).
There are constants rho, kappa > 0 (independent of i) with:
(i) On the orthogonal complement of the constants in
l^2(Vert Theta_i), d I - Adj_{Theta_i} >= rho I. Equivalently,
for every vertex subset W,
|boundary W| >= kappa * min(|W|, n_i - |W|), (2.2)
where boundary is the unoriented edge boundary.
(ii) n_i > 100, sigma_i > max(16, 20/c), lambda sigma_i > 3, and
1 < floor(lambda sigma_i) < floor(lambda sigma_{i+1}).
(iii) Every induced intrinsic ball of radius <= sigma_i / 16 is a TREE
with fewer than eta n_i = n_i / 100 vertices.
(iv) The k_i = floor(n_i/2) strictly increase and k_{i+1} - k_i -> inf.
PROOF. Step A: expanders with log girth. Start from a fixed-degree family
of connected rho-expander graphs of unbounded order and girth at least a
positive constant times log(order); the Lubotzky-Phillips-Sarnak
(Ramanujan) construction supplies such a family [LPS1988, Theorems 3.4,
4.1], and [Osajda2020, Sec. 2.4] records this form. Discard finitely many
small graphs so all are simple. All members share the gap rho > 0 off the
constants. Bipartiteness is harmless: an adjacency eigenvalue -d gives
Laplacian eigenvalue 2d, not another zero.
Step B: edge expansion from the spectral gap. For W of size w, the
centered indicator 1_W - (w/n)1 has squared norm w(n-w)/n and Dirichlet
energy equal to |boundary W|. Hence |boundary W| >= rho w(n-w)/n. Since
w(n-w)/n >= (1/2) min(w, n-w), take kappa = rho/2 to get (2.2).
Step C: bounded diameter/girth ratio. A ball holding at most half the
vertices has outside vertex boundary >= (kappa/d) times its own size, so
balls grow by a fixed factor until they hold more than half the graph.
Any two such big balls intersect, giving a uniform O(log n) diameter
bound. So the diameter/girth ratio is bounded.
Step D: small balls are trees. An induced ball of integer radius r with
2r+1 < sigma is a tree (an edge outside a shortest-path tree would close
a cycle of length <= 2r+1). Its order at r <= sigma/16 is at most
1 + d * sum_{j=0}^{floor(sigma/16)-1} (d-1)^j.
Step E: girth dominates growth. Regularity and the same tree property at
radius floor((sigma-2)/2) give the lower bound
n >= 1 + d * sum_{j=0}^{floor((sigma-2)/2)-1} (d-1)^j.
The ratio (Step D order)/(Step E order) tends to 0 as sigma grows, so
along a subsequence every ball of radius <= sigma/16 has fewer than
eta n vertices (iii). A further subsequence makes floor(lambda sigma_i)
strictly increasing (girth grows) while keeping (ii); the gaps
k_{i+1}-k_i grow without bound because n_i does (iv).
Step F: the labelling (the one genuinely external input). The sequence
has bounded degree, growing girth, bounded diameter/girth ratio, and
strictly increasing floors. Osajda's labelling theorem [Osajda2020,
Thm 2.7], applied to the ALREADY CHOSEN lambda, gives a reduced
labelling of this whole sequence over ONE finite alphabet S with NO
repeated long path word: the label word of a simple path of length
>= lambda sigma_i appears on no other simple path in the family; its
nonbacktracking-path conclusion includes the simple-path condition we
use. The associated graphical presentation has isometrically embedded
relators [Osajda2020, Lemma 3.1 and Thm 3.2], exactly the input Part 3
needs. QED
ORDER OF CHOICES (why the construction is not circular). The hyperbolicity
argument of Part 4 uses only the THRESHOLD lambda <= 1/12, never the exact
value of lambda: for every lambda below that threshold the filling bound
is the fixed "at most 12n faces of perimeter at most 4," so [BH1999] gives
ONE universal D_0. Thus D_0, then t = D_0+3 and L_0 = 2+2t+2D_0, are fixed
first. Only then do we pin down lambda, taking it below 1/24 and small
enough that 12(L_0+1)lambda <= 1/16 (2.1). The single alphabet S is fixed
after that by Osajda's theorem, and the prime p and the codes C_i after
that. The spectral constant epsilon = min(rho, c^2/(8d)) comes last. Each
row uses only earlier rows, so there is no circular dependence.
From now on the graphs, the single alphabet S, Gamma, its Cayley graph X,
and the coned graph X-hat are fixed.
================================================================
PART 3. THE CAYLEY GRAPH, THE COPIES, AND THE CONING
================================================================
Let S^{+-1} be the labelled oriented edge alphabet (labels and formal
inverses). The labelling is REDUCED: locally injective into the bouquet
on S (the outgoing labels at each vertex are distinct). We also have
0 < lambda <= 1/12, lambda sigma_i > 3 (3.1)
(the lambda < 1/24 of Part 2 implies this). We keep the uniqueness
property: the label word of a simple path in Theta_i of length
>= lambda sigma_i occurs on no other simple path in the family.
DEFINITION 3.1 (the group and its Cayley graph).
Gamma = < S | labels of all closed walks in all Theta_i >,
the group presented by declaring every closed-walk label word trivial.
Let X = Cay(Gamma, S): vertices Gamma, a directed edge h -> hs labelled s
for each s in S, reverse traversal labelled s^{-1}. We first treat X as
a FORMAL labelled graph, deterministic for reading words, before we know
loops or multiple edges are absent. Proposition 5.2 proves X is simple.
DEFINITION 3.2 (copies). Each Theta_i gives a based label map
phi_i : Theta_i -> X that is an ISOMETRIC EMBEDDING on vertices (Osajda:
relators are isometrically embedded). For g in Gamma write
P = P_{g,i} = g phi_i(Theta_i) ("copy")
with its source identification, and set sigma(P_{g,i}) = sigma_i. A copy
is any left translate of an embedded Theta_i. Write d_P for intrinsic
distance inside a copy P.
LEMMA 3.3 (distinct copies share no long simple path). Two different
indexed copies P, Q cannot share a simple edge path of length
>= lambda sigma(P) in either copy. In particular distinct index pairs
give distinct labelled subgraphs.
PROOF. Read a shared path in one direction in X. Its pullbacks to the
two source graphs are simple, since the vertex maps are injective. Label
uniqueness (Part 2 Step F) identifies the two source paths and hence the
two source graphs (same index). Equality at the first target vertex
identifies the translating elements, so the copies coincide as index
pairs. (The source graphs have simple paths longer than lambda sigma by
regularity and girth, so the threshold is never vacuous.) QED
THE CONED GRAPH X-hat. Adjoin to X a new "center" c_P for every copy P,
joined by a unit edge to EVERY vertex of P. All edges have length 1. The
result is X-hat: inside a copy you may travel between two of its vertices
the long way around X or the short way through c_P (length 2).
For a graph Y, Z_1(Y; F_2) is its ordinary FINITE-SUPPORT cycle space:
finite F_2-sums of edges with even incidence at every vertex. No
completed chain spaces are used.
================================================================
PART 4. HYPERBOLICITY OF THE CONED GRAPH
================================================================
The one hard geometric fact: X-hat is hyperbolic with a constant D_0
depending only on universal (coarse-filling) data, NOT on the particular
graphs, alphabet, or lambda. We prove it with the "adjustable sides"
diagram argument.
MOTIVATION. A closed edge walk in X-hat has two kinds of sides: generator
edges of X (fixed) and two-edge passages through a center (adjustable: we
may re-route the passage inside its copy). A closed walk lifting to a
trivial word has a van Kampen diagram in X whose bounded faces are
closed walks in individual copies. Minimizing the face count forces every
"arc" separating two different copies to be SHORT (below the girth of
either copy). Counting arcs gives a linear filling inequality, equivalent
to hyperbolicity.
DEFINITION 4.1 (diagrams with adjustable sides). A nonempty closed edge
walk in X-hat based at a group vertex is partitioned into m SIDES, each a
generator edge of X (fixed) or a two-edge passage through a center
(adjustable). INFLATE each center side to any walk between its endpoints
in its assigned copy (empty walk allowed when endpoints coincide);
generator sides stay fixed. The result is a closed walk in X.
For an inflated loop use a finite connected PLANE multigraph diagram whose
exterior contour maps to the loop and whose bounded-face contours are
closed walks in assigned copies. Such diagrams exist by van Kampen's
lemma: the presentation trivializes every source-graph closed walk, and
the group value at a face's start selects its translated copy. Contours
may repeat vertices and edges (essential: deleting a shared edge merges
faces, but other contacts persist as repeated vertices or bridges). The
merged bounded contour and each adjusted exterior side must remain walks
in their assigned copies with the prescribed endpoints.
An INCIDENCE is one occurrence of an edge on a contour; a bridge has two
incidences on the same contour. The exterior contour keeps its side
occurrences. Mark positions where sides end and the corresponding plane
vertices; there are AT MOST m distinct marked vertices. Only marked
POSITIONS are side breaks; revisiting a marked vertex need not end the
current side. Empty inflation allows an isolated marked vertex.
MINIMALITY. Minimize first the number F of bounded faces, then the edge
count, over all diagrams and adjustable walks with the prescribed side
data. SUPPRESS every unmarked degree-two vertex; the remaining edges are
ARCS, b of them, each keeping its original length. Loop arcs are allowed.
LEMMA 4.2 (short arcs and their count). In a minimal diagram:
(a) Every bounded face assigned to P has perimeter >= sigma(P), and
every arc incident to it has original length < lambda sigma(P).
(b) An arc with different copy assignments on its two incidences has
length < lambda sigma(Q) for either assignment Q. An arc with a
generator-side incidence has length 1.
(c) (1/lambda - 6) F <= 6m - 6, and b <= 6m. (4.1)
PROOF. (Consequences of minimality, incidences understood throughout.)
Face words are not freely trivial. The diagram identity writes the
exterior word as a product of conjugates of the face words, each used
once. Omitting a freely trivial factor and re-applying van Kampen gives a
diagram with fewer faces for the SAME exterior word; prescribed exterior
paths lift identically because X is deterministic. A closed walk in P
shorter than sigma(P) freely cancels to empty, so each face has
perimeter >= sigma(P).
No two faces with the SAME assignment share an edge. That edge is not a
bridge. Delete it and join the two contours along the deleted incidences;
every remaining step lies in the same copy, so the merged contour is an
admissible closed walk there, and exactly one bounded face disappears.
Other contacts may create repeated vertices or bridges, which our diagram
category allows. Contradiction to minimality of F.
No edge is shared by a bounded face and an ADJUSTABLE exterior side of
the SAME assignment: replace that occurrence in the side by the rest of
the bounded contour; the side is still a walk in that copy with the same
endpoints and the same prescribed exterior position, and one face
disappears.
No unmarked vertex of valence 1 (its leaf edge is an immediate backtrack
inside one adjustable exterior side; erasing it preserves assignments and
lowers the edge count).
No bridge whose two incidences belong to a bounded face. Remove the open
bridge edge; the exterior contour lies in one component. Keep that
component; discard the other component with the bridge. On the affected
bounded contour this replaces q e gamma e^{-1} q' by q q' (e the
bridge, gamma the excursion around the discarded component). The
retained walk stays closed and in the assigned copy (no free triviality
of gamma needed), the exterior and side data are unchanged, F does not
increase, and the edge count drops.
The incidence type on either side of a compressed arc is constant (a
bounded face, or an exterior side of the decomposition; exterior-side
changes happen at marked vertices and are not suppressed). Hence an arc
with a generator-side incidence has length 1, and a face-incident arc has
either a generator incidence on the other side or a DIFFERENT assignment.
Now take an arc with different assignments and length > 2. Its word is
freely reduced: a consecutive cancelling pair has an unmarked bivalent
midpoint (true even for a loop arc, whose internal vertices are distinct);
contracting that length-two tree identifies two endpoints with the same
group value and preserves assignments, F, and side data while lowering
the edge count. If the arc had length >= lambda sigma(Q) for one
assignment Q, its first ceil(lambda sigma(Q)) edges form a
nonbacktracking path shorter than girth, simple in Q and hence in the
other source by vertex embeddings, contradicting Lemma 3.3. Arcs of
length <= 2 are already below threshold. This proves (a) and (b).
Counting. There is a marked vertex, so suppression leaves no exceptional
unmarked circle component; every surviving unmarked vertex has valence
>= 3. With v_D the compressed vertex count,
v_D - b + F = 1, 2b >= 3(v_D - m), hence b <= 3F + 3m - 3.
Each bounded face has >= 1/lambda arc incidences (with multiplicity),
total <= 2b, so F/lambda <= 2b <= 6F + 6m - 6, the first inequality in
(4.1). Since lambda <= 1/12 it gives F <= m - 1 and then b <= 6m. Trees
and the isolated marked vertex are covered by the same formulas. QED
PROPOSITION 4.3 (uniform thinness; existence of D_0). There is an integer
D_0, independent of all the data, such that every vertex on a geodesic
triangle in X-hat is within D_0 of a vertex on another side.
PROOF. Step 1 (linear short filling). Every edge loop of length n > 0 in
X-hat has a filling with at most 12n relation faces of perimeter at most
4. For its m <= n sides take the minimal compressed diagram of Lemma 4.2.
Choose one STANDARD path per arc, consistent under reversal: keep the
unit edge if the arc has a generator-side incidence; otherwise an
assignment incidence contains both endpoints, so use the two-edge path
through that copy's center.
Around a bounded face assigned to P, replace each standard path by the
two-edge path through c_P: each replacement is one closed walk of length
<= 4, and the contour then freely contracts through that single center.
On an exterior adjustable side do the same through its prescribed center;
consecutive pairs reduce to the single required cone passage. An empty
inflated side gets a free backtrack; generator sides are unchanged. Total
cost = number of bounded-face and exterior adjustable-side incidences
<= 2b <= 12m <= 12n.
Step 2 (van Kampen bookkeeping). The plane-diagram identity writes the
exterior contour as a product of conjugates of the bounded contours, one
per face, up to free edge cancellations and inverses. Substituting the
fillings gives a product of short relations; conversely van Kampen
reconstructs a plane diagram over the graph (inverse-edge folds identify
only vertices with the same target value).
Step 3 (fillings give thin triangles). The at-most-12n-perimeter-4 bound
is the coarse filling hypothesis of [BH1999, III.H.2.1 and III.H.2.9]
with universal constants. Subdivide a rectifiable loop of length ell into
O(ell+1) arcs of length <= 1, pick nearby vertices at distance <= 1, join
successive ones by paths of <= 3 edges, and fill that edge loop as above.
In a small-face diagram with f faces and exterior length n,
2E = n + sum_faces |boundary| <= n + 4f, and V <= E + 1. Thicken the
finite plane graph to a regular neighborhood (vertex disks, edge strips,
caps over bounded contours); the image is a disk mapping vertices to
vertices, strips to edges, each cap to bounded image perimeter. Triangulate
at cost O(E + V + f), each triangle a map of uniformly bounded image
diameter (coarse fillings need no continuity on interiors). A collar with
O(n) triangles restores the prescribed exterior parameterization.
Constant loops and empty walks are filled by a bounded number of
triangles.
The constants depend only on the coarse-filling constants, not on the
graphs, alphabet, or lambda. A connected unit-edge graph is geodesic
(regardless of local finiteness), so [BH1999] gives a hyperbolicity
constant depending only on these universal constants. Rounding up to an
integer and moving points on triangle sides to vertices gives D_0. QED
================================================================
PART 5. AVOIDANCE AND SIMULTANEOUS OVERLAPS
================================================================
Turn thinness into the two estimates the later algebra needs: an
avoidance bound (a path in a copy that dodges the center is short in
intrinsic distance) and a SIMULTANEOUS overlap bound (all far-or-equal
copies meet a fixed copy in ONE short tree ball).
Set, once and for all,
t = D_0 + 3, L_0 = 2 + 2t + 2 D_0. (5.1)
LEMMA 5.1 (avoidance). If an edge path of length L joins x, a in a copy P
while avoiding c_P, then
d_P(x, a) <= 12 (L + 1) lambda sigma(P). (5.2)
PROOF. Close the avoiding path by one designated side through c_P; then
m <= L + 1 and no other side has center c_P. Take a minimal diagram
(Lemma 4.2). An arc incidence on the designated side faces either a
bounded face of different assignment or an exterior incidence across a
bridge. In the bridge case the other incidence cannot also lie in the
designated side: if it did, the interval from that bridge's first
traversal to its retraversal in the LINEAR side walk would contain no
prescribed side break; excising the visited component and the bridge
shortens only the designated side, preserving endpoints, copy
assignment, and F, contradicting minimality (this concerns marked
positions; extra visits to a marked vertex insert no side break). So
every designated-side incidence has length < lambda sigma(P) (Lemma
4.2(b), or a generator incidence). There are at most 2b such
occurrences, so the adjusted walk inside P has length <= 12(L+1) lambda
sigma(P). Empty walks (including x = a) satisfy it. QED
PROPOSITION 5.2 (X is simple). If lambda < 1/12 then X is simple: no
nonfreely-trivial closed label word in X has length <= 2.
PROOF. Consider a closed word of one or two generator sides. Strict
lambda < 1/12 in (4.1) forces F = 0, and the exterior contour of a tree
is freely trivial. This excludes identity generators, unintended
coincidences of generator edges, and generator involutions. QED
LEMMA 5.3 (simultaneous overlap bound). Let R be a group vertex or a
center of X-hat, and A = c_P different from R. Choose one geodesic from A
to R; its first group vertex is some a in P. Then every vertex shared by
P and ANY different copy Q whose center satisfies d(c_Q, R) <= d(c_P, R)
lies in the intrinsic induced ball B_P about a of radius
r_P = floor( 12 (L_0 + 1) lambda sigma(P) ). (5.3)
PROOF. Take such a Q and a shared vertex x; write B = c_Q. The path
A, x, B is a length-two geodesic. A geodesic from B to R avoids A (else
it would have length 2 + d(A,R) > d(A,R)). All distances are in X-hat.
Case d(A,R) >= t. Let z be the vertex at distance t along the chosen
A-to-R geodesic; it is at distance >= t - 2 > D_0 from every vertex of
A, x, B. Thinness supplies a vertex y on the B-to-R side with
d(y, z) <= D_0. A shortest bridge from y to z avoids A (because
d(A,z) = t > D_0). Follow x, B, y, z, a using that bridge and the
indicated geodesic segments. This walk avoids A and has length at most
1 + (2 + t + D_0) + D_0 + (t - 1) = L_0.
Case d(A,R) < t. Use the walk from x through B to R and back to a; it
avoids A and has length at most 1 + d(B,R) + d(A,R) - 1 <= 2 d(A,R) <=
L_0.
Apply (5.2): d_P(x, a) <= 12 (L_0 + 1) lambda sigma(P). Intrinsic vertex
distances are integers, so taking the floor gives r_P. The chosen a does
NOT depend on Q or x, which is the "simultaneous" part. QED
CONSEQUENCE. Every ball B_P from Lemma 5.3 is an induced subtree of P
with r_P <= sigma(P)/16 and |B_P| < eta |P| = |P|/100. Indeed, by (2.1),
12(L_0+1)lambda <= 1/16 gives r_P <= sigma(P)/16, and Proposition
2.1(iii) makes a radius-<= sigma/16 induced ball a tree with fewer than
eta |P| vertices. (5.4)
================================================================
PART 6. CODES OVER F_p
================================================================
We attach to each expander a linear code over a finite field F_p with
two distance guarantees at once: the code contains the prescribed
"port pattern" and neither the code nor its dual contains a sparse
nonzero vector. Primal distance forces many label discrepancies on any
copy that is not in the prescribed state; dual distance keeps the module
relations from cancelling across copies in Part 7.
Choose a prime p > 2 large enough that the d-element subsets of S^{+-1}
inject into F_p^{x}: J |-> a_J. This is possible because there are only
finitely many such subsets. Define the DECODER, for a in F_p,
J(a) = J if a = a_J for some d-element subset J,
J(a) = empty otherwise.
At a vertex v of Theta_i, let J_i(v) be its set of outgoing labels; by
reducedness and d-regularity |J_i(v)| = d. The PROTOTYPE PATTERN is
w_i(v) = a_{J_i(v)}; every coordinate of w_i is nonzero. Transporting
coordinates to a copy P = P_{g,i} gives J_P(h) and w_P(h), with
J(w_P(h)) = J_P(h) for every h in P.
On the finite coordinate space F_p^E use the nondegenerate bilinear form
<z, z'> = sum_{v in E} z(v) z'(v) in F_p,
and let supp z = {v in E : z(v) is nonzero}.
LEMMA 6.1 (codes with both distances). For every i there is a
k_i-dimensional subspace C_i of F_p^{Vert(Theta_i)} such that w_i is in
C_i and
|supp z| > eta n_i for every nonzero z in C_i or in C_i^perp.
(6.1)
PROOF (Gilbert-Varshamov counting). Write n = n_i, k = k_i, w = w_i, and
choose C uniformly among the k-dimensional subspaces containing <w>. For
a fixed z outside <w>, the quotient by <w> gives
Pr(z in C) = (p^{k-1}-1)/(p^{n-1}-1) < p^{k-n}.
Reason: a uniform a-dimensional subspace of an N-dimensional space
contains a fixed nonzero vector with probability (p^a-1)/(p^N-1). Nonzero
vectors of <w> have full support, so they never count as sparse.
Orthogonal complementation is a bijection of this sample space to the
(n-k)-dimensional subspaces of the (n-1)-dimensional space w^perp, so for
a fixed nonzero z,
Pr(z in C^perp) = (p^{n-k}-1)/(p^{n-1}-1) < p^{1-k} if z in w^perp,
Pr(z in C^perp) = 0 otherwise.
(Valid even when <w,w> = 0: the form on the ambient space is what
matters.)
Now count low-weight vectors. Put r = floor(eta n) and q = eta/(1-eta) <
1. The binomial theorem gives
q^r sum_{j=0}^r binom(n,j) <= sum_{j=0}^r binom(n,j) q^j <= (1+q)^n,
hence sum_{j=0}^r binom(n,j) <= q^{-eta n} (1+q)^n = 2^{H_2(eta) n} with
H_2(t) = -t log_2 t - (1-t) log_2 (1-t). Since H_2(1/100) < 9/100 and
p >= 2,
|{z : |supp z| <= r}| = sum_{j=0}^r binom(n,j)(p-1)^j
<= p^{eta n} 2^{H_2(eta) n} < p^{n/4}.
Every nonzero vector in this ball lies outside <w>. By the two
probabilities above and the union bound, the probability that C OR
C^perp contains such a vector is less than
p^{n/4} (p^{k-n} + p^{1-k}) <= p^{-n/4} + p^{3/2 - n/4} < 1,
because k = floor(n/2) and n > 100. A suitable C therefore exists. QED
Choose C_i for each i. Transporting coordinates defines C_P and
C_P^perp on every copy P; the transport preserves the form, so these are
orthogonal complements in F_p^P.
================================================================
PART 7. THE MODULE EXTENSION AND ITS EXACT RELATIONS
================================================================
We impose the dual code relations on every translated copy and check
that the result creates EXACTLY the prescribed relations on each copy:
no cross-copy cancellation survives.
Let F_p^{(Gamma)} be the finite-support coordinate space with basis
(e_h)_{h in Gamma}, on which Gamma acts by g e_h = e_{gh}. Regard F_p^P
as its coordinate subspace by extension by zero, and put
W_0 = sum over g in Gamma, i >= 1 of C_{P_{g,i}}^perp
contained in F_p^{(Gamma)}. (7.1)
Every vector of W_0 is a finite sum of terms, and left translation
carries C_{P_{g,i}}^perp to C_{P_{hg,i}}^perp, so W_0 is
Gamma-invariant.
LEMMA 7.1 (exact coordinate relations). For every copy P,
W_0 ∩ F_p^P = C_P^perp. (7.2)
PROOF. The inclusion from the right follows from (7.1). For the other,
let z = sum_{Q in F} z_Q with 0 =/= z_Q in C_Q^perp and F finite and
distinct, and suppose z lands in F_p^P. Root the coned graph at R = c_P,
and choose Q in F minus {P} whose center is farthest from R. Every
other copy in F, and also P, has center at distance from R at most
d(c_Q, R). Lemma 5.3 puts all their vertex intersections with Q inside
one intrinsic ball B_Q, and the Consequence (5.4) gives
|B_Q| < eta n_j, with j the type of Q, while Lemma 6.1 gives
|supp z_Q| > eta n_j. Choose h in supp z_Q minus B_Q. Then h lies in no
other copy in F and not in P, so the h-coordinate of z is z_Q(h) =/= 0,
contradicting z in F_p^P. So the expression has no term away from P,
hence z is in C_P^perp. QED
DEFINITION 7.2 (the group). Put
V = F_p^{(Gamma)} / W_0, v_* = [e_1] in V,
with the induced Gamma-action, and
G_sur = V rtimes Gamma,
with multiplication (v,g)(v',g') = (v + g v', g g').
For h in Gamma, h v_* = [e_h]. G_sur is countable. It is also finitely
generated: the additive group of V is generated by the Gamma-orbit of
v_*, since those are the images of the coordinate basis and scalar
multiples are repeated additions, and in the semidirect product
(0,h)(v_*,1)(0,h)^{-1} = (h v_*, 1).
So (v_*,1) together with the lifts (0,s), s in S, generates G_sur. This
proves the "finitely generated" part of the main theorem.
================================================================
PART 8. THE COMPACT DUAL AND ITS UNIFORM MARGINALS
================================================================
Write Omega = Hom_{F_p}(V, F_p) for the compact dual, with the character
v |-> exp(2 pi i omega(v)/p) and the topology of pointwise convergence
(equivalently the subspace topology from F_p^V). It is a closed subgroup
of that compact product, hence a compact abelian group; let m_Omega be
its Haar probability. For omega in Omega and h in Gamma set
X_omega(h) = omega(h v_*).
(X_omega is a coordinate PATTERN, not the Cayley graph X.)
LEMMA 8.1 (uniform pattern marginals). For a copy P = P_{g,i} the
restriction map
res_P : Omega -> F_p^P, omega |-> X_omega|_P,
has image exactly C_P; its pushforward of m_Omega is the uniform measure
on C_P. In particular, for every a in C_P,
m_Omega{omega : X_omega|_P = a} = p^{-k_i}, (8.1)
and this applies to a = w_P.
PROOF. Let q_P : F_p^P -> V be the restriction of the quotient map.
Lemma 7.1 gives ker q_P = C_P^perp. For z in F_p^P and omega in Omega,
omega(q_P(z)) = sum_{h in P} z(h) omega(h v_*) = <z, X_omega|_P>.
So X_omega|_P annihilates C_P^perp, hence lies in (C_P^perp)^perp = C_P.
Conversely, for a in C_P the rule q_P(z) |-> <z,a> is a well-defined
linear functional on q_P(F_p^P); extend a basis of that subspace to a
basis of V, extend the functional linearly, and get omega in Omega with
X_omega|_P = a. No continuity is lost because V is discrete.
Finally res_P is a continuous homomorphism of compact groups onto the
finite group C_P, so its pushforward of Haar probability is
translation-invariant and hence uniform on C_P. Since |C_P| = p^{k_i},
formula (8.1) follows. QED
REMARK. Lemma 8.1 fixes each INDIVIDUAL copy marginal and asserts no
independence of different copies. What makes these marginals survive the
relations imposed on every translated copy is the exact intersection
(7.2), which in turn rests on the simultaneous containment of all
relevant overlaps in one ball.
================================================================
PART 9. THE OPERATOR AND THE ENABLED GRAPH
================================================================
Fourier transformation on the discrete abelian group V identifies
C*_r(V) with C(Omega). Regular induction of the left regular
representation of Gamma then gives
C*_r(G_sur) = C*_r(V rtimes Gamma) = C(Omega) rtimes_{alpha,r} Gamma,
(9.1)
where alpha_g(f)(omega) = f(g^{-1} omega), and u_g is the canonical
unitary for g in Gamma (so u_g f u_g^* = alpha_g(f)).
For omega in Omega, the representation induced by evaluation at omega
acts on l^2(Gamma) by
(pi_omega(f) xi)(h) = f(h^{-1} omega) xi(h),
(pi_omega(u_g) xi)(h) = xi(hg). (9.2)
The direct sum of all pi_omega is faithful, so the norm in (9.1) is the
sup over this family.
The canonical trace of G_sur is, for b in C*_r(G_sur),
tau_{G_sur}(b) = integral_Omega <pi_omega(b) delta_1, delta_1> dm_Omega(omega).
(9.3)
For a finite crossed-product sum b = sum_g F_g u_g the diagonal
coefficient at 1 is F_1(omega), whose Haar integral is the canonical
group trace under Fourier transformation; density and uniform
convergence of diagonal coefficients extend this to all b, and the
integrand is continuous.
For s in S^{+-1} let f_s in C(Omega) be the clopen-set projection
f_s(omega) = 1 if s is in J(omega(v_*)),
f_s(omega) = 0 otherwise.
Define the operator
T = d I - sum_{s in S^{+-1}} f_s u_s f_{s^{-1}} in C*_r(G_sur).
(9.4)
It is self-adjoint (adjoints interchange s and s^{-1}); each f_s is a
function of the single finite-valued coordinate omega(v_*), hence a
finite Fourier sum of characters of Omega, so T lies in the group
algebra of (9.1).
DEFINITION 9.1 (enabled graph). For omega in Omega, the enabled graph
G_omega has vertex set Gamma; at h the allowed outgoing labels are
J(X_omega(h)). The Cayley edge h -- hs is ENABLED precisely when s is
allowed at h and s^{-1} is allowed at hs. A copy P is VALID for omega if
X_omega|_P = w_P. Let U_omega be the complement in Gamma of the union
of all valid copies.
G_omega is undirected and simple, degrees at most d (the decoder
returns either a d-element set or empty). Since
(h^{-1} omega)(v_*) = X_omega(h), formula (9.2) gives
pi_omega(T) = d I - Adj(G_omega). (9.5)
The degree bound gives ||Adj(G_omega)|| <= d, so pi_omega(T) is bounded,
positive, and at most 2d I.
On a valid copy P the allowed port sets are exactly J_P(h); every copy
edge is enabled and the d incidences at each vertex exhaust the allowed
labels. Since P is connected, it is a WHOLE ISOLATED COMPONENT of
G_omega with exactly its copy edges. Distinct valid copies are disjoint:
two valid copies meeting at a vertex would be the entire same component;
if their indices differed, root X-hat at that shared vertex, so both
centers are at distance one, and Lemma 5.3 would place their whole common
vertex set inside a ball of the first copy with fewer than eta|P| < |P|
vertices, impossible since the copy itself has |P| vertices. Hence no
enabled edge joins U_omega to its complement.
================================================================
PART 10. DEFICITS: LOCAL INSIDE A COPY, GLOBAL BY CYCLE RANK
================================================================
For B a subset of a copy P, let e_{omega,P}(B) be the number of enabled
copy edges with both endpoints in B. The two deficits below are the
combinatorial heart; the spectral gap is a routine consequence.
LEMMA 10.1 (deficit inside an invalid copy). If P is invalid for omega,
then EVERY subset B of P satisfies
d|B| - 2 e_{omega,P}(B) >= c|B|. (10.1)
PROOF. Write n = |P| and M = {h in P : X_omega(h) =/= w_P(h)}. By Lemma
8.1, X_omega|_P lies in C_P, and w_P lies in C_P by Lemma 6.1; being
invalid, their difference is nonzero, so the primal distance (6.1) gives
|M| > eta n. At a vertex of M the decoder returns either the empty set
or a d-element set different from the prescribed J_P(h); in either case
at least one prescribed incidence is missing. So, with b = |B|,
d b - 2 e_{omega,P}(B) >= |M ∩ B| and >= |boundary_P B|.
If n - b >= eta n/2, then min(b, n-b) >= (eta/2) b, and the expansion
(2.2) gives |boundary_P B| >= kappa min(b, n-b) >= (eta kappa/2) b >=
c b. Otherwise n - b < eta n/2, so |M ∩ B| >= |M| - (n-b) > eta n/2 >=
(eta/2) b >= c b. We used c = (eta/2) min(kappa, 1). The case B empty is
included. QED
LEMMA 10.2 (deficit on the remaining graph). For every omega in Omega and
every finite Y contained in U_omega, let A be the graph on Y with all its
internal enabled edges. Then
d|Y| - 2|E(A)| >= (c/2)|Y|. (10.2)
PROOF. The local deficits cannot be added over copies, because copies and
their enabled edges overlap. Instead bound the cycle rank
b_1(A) = dim Z_1(A; F_2). Euler's identity |E(A)| = |Y| - #pi_0(A) +
b_1(A) (valid for isolated vertices and the empty graph) shows that
(10.2) follows from
b_1(A) <= (d/2 - 1 - c/4)|Y|. (10.3)
Cycle coordinates. View L = Z_1(A; F_2) inside Z_1(X; F_2) = direct sum
over P of Z_1(P; F_2) (Lemma 11.0 below). Coordinate maps pr_P are those
of the algebraic decomposition, NOT restrictions of edge chains to P: a
projected cycle may contain edges absent from the original cycle. Only
finitely many coordinates are nonzero on L (take a finite basis and union
its finite coordinate supports). Call them P_1, ..., P_N, ordered by
NONINCREASING distance of centers from a fixed group-vertex root R in
X-hat; break ties arbitrarily. For 1 <= j <= N+1 set
L_j = {z in L : pr_{P_i} z = 0 for i < j}.
Then L_1 = L, L_{N+1} = 0, and rank-nullity gives
dim L = sum_{j=1}^N dim pr_{P_j}(L_j). (10.4)
Localizing a projected cycle. Fix j and P = P_j; take the induced tree
ball B_P of Lemma 5.3 for this root. It contains every vertex of P shared
with any remaining copy P_{j+1}, ..., P_N. Let D_P = B_P union (A ∩ P),
where A ∩ P has vertex set Y ∩ P and the edges common to A and P. CLAIM:
pr_P(L_j) is contained in Z_1(D_P; F_2). (10.5)
Indeed let edge e occur with nonzero coefficient in pr_P z, z in L_j. If
e is not in A, its coefficient must cancel with one from another
remaining copy (the total chain z is supported in A); both endpoints of e
are then shared with that copy and lie in B_P, and inducedness of the
ball puts e itself in B_P. The remaining edges of pr_P z lie in A ∩ P.
The projection is already a cycle in P, so (10.5) holds. Hence
dim pr_P(L_j) <= b_1(D_P). (10.6)
Bounding exterior cycles and attachments. If P is valid then Y ∩ P is
empty, so D_P = B_P is a tree and the right side of (10.6) is 0. Suppose
P is invalid; write sigma = sigma(P) and r for the radius of B_P, so
r <= sigma/16 and sigma > max(16, 20/c).
Consider the connected components C of D_P after deleting the vertices of
B_P; their vertex sets partition (Y ∩ P) minus B_P. Let l_C be the number
of edges joining C to B_P. Since B_P is a connected tree, the Euler
formula for graph cycle rank gives
b_1(D_P) = sum_C (b_1(C) + max(0, l_C - 1)). (10.7)
Each edge of C is an enabled copy edge and all enabled copy edges between
its vertices occur in C (definition of A). A cyclic C has at least sigma
vertices; applying Lemma 10.1 to its vertex set and using connectedness,
b_1(C) = |E(C)| - |C| + 1
<= (d/2 - 1 - c/2 + 1/sigma)|C|. (10.8)
The same bound holds for an acyclic C (then b_1(C) = 0 and the
coefficient is positive); here d >= 3 and 0 < c <= eta/2 = 1/200.
Extra rank from attachments to the tree. Take two distinct attachment
edges, endpoints v, v' in C and b, b' in B_P. A shortest path in C from v
to v', the two edges, and the tree path from b' to b form a simple cycle
in P (if one pair of endpoints agrees, use a zero-length path; both
pairs cannot agree since P is simple). The tree path has length at most
2r because both endpoints lie in the radius-r ball. Girth gives
dist_C(v, v') >= sigma - 2r - 2 > sigma/2. (10.9)
In particular all attachment endpoints in C are distinct. If l_C >= 2,
the balls in C of radius floor(sigma/4) centered at these endpoints are
pairwise disjoint by (10.9), and each has at least floor(sigma/4)+1 >
sigma/4 vertices. Thus
max(0, l_C - 1) <= (4/sigma)|C|. (10.10)
For l_C = 0 or 1 the same inequality is immediate. Combining (10.7),
(10.8), (10.10), and 5/sigma < c/4,
b_1(D_P) <= (d/2 - 1 - c/2 + 5/sigma)|(Y ∩ P) minus B_P|
<= (d/2 - 1 - c/4)|(Y ∩ P) minus B_P|. (10.11)
As noted, it holds also for a valid copy (both sides vanish).
Summing over disjoint vertex sets. The exterior vertex sets charged in
(10.11) are pairwise disjoint: if j < k then the center of P_k is no
farther from R than that of P_j, so Lemma 5.3 gives P_j ∩ P_k inside
B_{P_j}, hence (Y ∩ P_j) minus B_{P_j} misses all of P_k including its
charged exterior. The nonstrict comparison handles ties. Summing (10.6)
and (10.11) in (10.4) yields (10.3). Substituting into Euler's identity
proves (10.2). QED
LEMMA 11.0 (direct sum of copy cycles). The copy inclusions give an
internal algebraic direct sum
Z_1(X; F_2) = direct sum over P of Z_1(P; F_2). (11.0)
PROOF. Spanning. Every finite cycle is a sum of closed walks. Each such
walk is trivial in Gamma, so a finite van Kampen diagram over the
presentation writes its edge chain, modulo two, as a sum of bounded-face
walk chains, each in a copy.
Independence. Suppose nonzero cycles in distinct copies sum to zero.
Choose a group-vertex root R and a participating copy P whose center is
farthest from R. Lemma 5.3 puts all its intersections with the other
participating copies in one ball B_P, which (5.4) makes an induced tree.
Every nonzero edge coefficient of its cycle must cancel against an edge
of another copy; both endpoints then lie in B_P, and inducedness puts the
edge itself in that tree, where no nonzero finite cycle exists. QED
================================================================
PART 11. UNIFORM SPECTRAL GAP AND THE PROJECTION
================================================================
The passage from finite-set expansion to a spectral estimate is the
discrete Cheeger argument [Dodziuk 1984; Alon-Milman 1985]. We spell out
the coarea and Cauchy-Schwarz bookkeeping to account for the vertex
potential d - deg_omega(x) and the unnormalized Laplacian.
LEMMA 11.1 (uniform spectral gap). For every omega in Omega, the
restriction of pi_omega(T) to l^2(U_omega) satisfies
pi_omega(T) restricted to l^2(U_omega) >= (c^2/(8d)) I. (11.1)
Put epsilon = min(rho, c^2/(8d)) > 0. Then
spec(pi_omega(T)) is contained in {0} union [epsilon, 2d], (11.2)
and the kernel is exactly the Hilbert direct sum of the one-dimensional
spaces of constants on the valid copies.
PROOF. On U_omega let q(x) = d - deg_omega(x) >= 0. There are no enabled
edges from U_omega to its complement. For finitely supported complex xi on
U_omega,
E(xi) := <pi_omega(T) xi, xi>
= sum over enabled edges {x,y} on U_omega of |xi(x)-xi(y)|^2
+ sum over x in U_omega of q(x)|xi(x)|^2. (11.3)
All edge sums below are over enabled edges on U_omega. For a finite set
F contained in U_omega,
d|F| - 2|E(G_omega restricted to F)| = |boundary F| + sum_{x in F} q(x).
Set s(x) = |xi(x)|. Apply Lemma 10.2 to the finite level sets
{x : s(x)^2 > t} and integrate in t >= 0; layer cake gives
(c/2)||xi||^2 <= sum_{x,y} |s(x)^2 - s(y)^2| + sum_x q(x) s(x)^2.
Cauchy-Schwarz applied together to the edge and vertex-potential terms:
(c/2)||xi||^2
<= ( sum_{x,y} |s(x)-s(y)|^2 + sum_x q(x) s(x)^2 )^{1/2}
( sum_{x,y} (s(x)+s(y))^2 + sum_x q(x) s(x)^2 )^{1/2}
<= E(xi)^{1/2} sqrt(2d) ||xi||.
First factor: | |xi(x)| - |xi(y)| | <= |xi(x) - xi(y)|. Second factor:
sum_{x,y} (s(x)+s(y))^2 + sum_x q(x) s(x)^2
<= sum_x (2 deg_omega(x) + q(x)) s(x)^2 <= 2d ||xi||^2.
For xi nonzero, divide and square: E(xi) >= (c^2/(8d))||xi||^2; the zero
vector obeys it too. Finite-support vectors are dense and the operator is
bounded, so (11.1) holds on all of l^2(U_omega) (vacuous if U_omega is
empty).
On each valid copy P of type i the operator is the graph Laplacian
d I - Adj(P): kernel = constants, and at least rho I off the constants by
Proposition 2.1(i). Valid copies are disjoint isolated components, so
decomposing l^2(Gamma) into their component spaces and l^2(U_omega) shows
the full kernel is exactly the direct sum of C 1_P over valid P and the
operator is at least epsilon I off it. This survives infinitely many valid
components because both bounds are uniform. The earlier bound
pi_omega(T) <= 2d I gives (11.2). QED
PROPOSITION 11.2 (a projection in the reduced group algebra). There is a
projection b_0 in C*_r(G_sur) such that, for every omega, pi_omega(b_0) is
the orthogonal projection onto ker pi_omega(T): projection onto constants
on each valid copy and zero on l^2(U_omega). In particular
<pi_omega(b_0) delta_1, delta_1>
= 1/n_i if 1 lies in a valid copy of type i,
= 0 if 1 lies in U_omega, (11.4)
and the valid copy in the first case is unique.
PROOF. Take the faithful direct sum of the pi_omega. Lemma 11.1 gives the
same epsilon in every summand, so the spectrum of this sum on T sits in
{0} union [epsilon, 2d]. Faithfulness preserves the spectrum of the
self-adjoint T in C*_r(G_sur), hence
spec_{C*_r(G_sur)}(T) is contained in {0} union [epsilon, 2d].
Pick a continuous real function chi on [0, 2d] with chi(0) = 1 and chi = 0
on [epsilon, 2d], and set b_0 = chi(T) by continuous functional calculus.
Since chi^2 = chi on the spectrum, b_0 is a projection in C*_r(G_sur). This
is ORDINARY C*-algebra functional calculus, with no passage to a von
Neumann algebra.
Functional calculus commutes with each pi_omega, so the kernel description
in Lemma 11.1 identifies pi_omega(b_0) as claimed, also when there are no
valid copies or infinitely many. On a valid copy of type i the normalized
constant vector is n_i^{-1/2} 1_P, so the diagonal of its rank-one
projection at any vertex is 1/n_i; at vertices of U_omega it is 0.
Disjointness of valid copies gives uniqueness and (11.4). QED
================================================================
PART 12. THE TRACE IS IRRATIONAL
================================================================
PROPOSITION 12.1 (irrational trace). The projection b_0 satisfies
tau_{G_sur}(b_0) = sum_{i>=1} p^{-k_i}, an irrational number in
[0,1). (12.1)
PROOF. Fix omega. The diagonal entry of pi_omega(b_0) at the identity is
0 unless the identity lies in a valid copy; if it does, the copy is unique
with some type i, and projection onto the constants on that component has
diagonal entry 1/n_i.
Exactly n_i indexed copies of type i contain the identity: the translating
elements are g = phi_i(v)^{-1} for the n_i distinct embedded vertices v of
Theta_i, and Lemma 3.3 makes these distinct copies. By Lemma 8.1 each is
valid on an event of Haar measure p^{-k_i}. Validity events for copies
containing the identity are mutually exclusive, including across types. By
(9.3) and nonnegative countable summation,
tau_{G_sur}(b_0) = sum_{i>=1} n_i (1/n_i) p^{-k_i} = sum_{i>=1} p^{-k_i}.
No independence is needed.
Since p > 2 and the positive integers k_i strictly increase, this number
lies in [0,1) and has base-p digits equal to one exactly at positions k_i
and zero elsewhere (no eventually-(p-1) expansion ambiguity). There are
infinitely many ones, but their successive gaps tend to infinity by
Proposition 2.1(iv), so the digits are not eventually periodic. Rational
numbers have eventually periodic base-p expansions; hence (12.1) is
irrational. QED
================================================================
PART 13. CONCLUSION: ASSEMBLY FAILS
================================================================
PROOF OF THE MAIN THEOREM. Part 7 gives a finitely generated discrete
group G_sur; Proposition 11.2 gives a genuine projection b_0 in
C*_r(G_sur); Proposition 12.1 gives that tau_{G_sur}(b_0) is irrational,
so in particular [b_0] is a nonzero integral K_0-class. By Lemma 1.1 every
class in the image of the proper-action assembly map has rational trace.
Hence [b_0] is outside mu_0^{G_sur}(K_0^{G_sur}(E~G_sur)). This is a
failure of surjectivity in degree zero, and it refutes the universal
coefficient-free Baum-Connes assertion. QED
The obstruction survives rationalization. Extend the canonical trace to
K_0(C*_r(G_sur)) tensor Q by x tensor q |-> q tau_{G_sur*}(x). Every
element in the image of mu_0 tensor id_Q has rational trace, as a finite
rational combination of assembly classes. The trace of [b_0] tensor 1 stays
irrational, so that class is outside the rationalized assembly image too.
REMARK (how this sits with the known positive results). The algebra
C(Omega) rtimes_r Gamma was used only to REPRESENT C*_r(V rtimes Gamma)
and to analyze T; the assembly map obstructed is the coefficient-FREE
reduced map for G_sur itself, with the full proper classifying space.
There is no conflict with the positive theorems, and in fact they FORCE
structural facts about G_sur. Amenable groups satisfy assembly with
coefficients (Higson-Kasparov) and hyperbolic groups satisfy assembly
coefficient-free (Mineyev-Yu); so the failure of assembly for G_sur
proves that G_sur is neither amenable nor hyperbolic. The genuine new
content beyond the earlier von Neumann results (Austin;
Pichot-Schick-Zuk; Grabowski) is exactly the C*-algebra distinction:
those produced projections in group von Neumann algebras, which need not
lie in the reduced group C*-algebra; here the uniform spectral gap
places the projection inside C*_r(G_sur).
END OF PROOF.