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THE l^1-BASS CONJECTURE FOR DISCRETE GROUPS
A complete, self-contained account
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Notation at a glance:
C complex numbers
Z integers
G a discrete group
e the identity element of G (also of a ring)
l^1(G) the complex group algebra of G with the l^1 norm
a, b, c elements of l^1(G), that is, functions G -> C
a(g) the value of a at g
a * b convolution of a and b
e_p an idempotent (p^2 = p) element of a matrix algebra
[g] the conjugacy class of g
C(G) the set of all conjugacy classes of G
trace_C the Hattori-Stallings trace at class C
K_0(A) the Grothendieck group of finitely generated projective
right A-modules
Sections 1-8 give the setup and state the main theorem. Section 9 is a
one-page proof strategy. Sections 10-16 carry it out, one step per
section, matching the items in section 9. Appendix A is a glossary.
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0. THE PROBLEM IN ONE PARAGRAPH
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A group ring is a way to turn a group into an algebraic object: we take
all finite (or summable) formal combinations of group elements. Over it,
a projective module and its K-theory class are abstractly the same data
as "an idempotent matrix with entries in the group ring". Since the
1960s, people have studied the trace of such a matrix because it gives
a computable invariant of the module. Hattori and Stallings showed how
to push the ordinary matrix trace down to a sum over conjugacy classes
of the group; this is the "Hattori-Stallings trace".
The l^1-Bass conjecture, in its cleanest form, says that for the complex
group l^1-algebra of any discrete group, this trace is supported on
finitely many conjugacy classes of finite-order elements. That is:
* it vanishes on every infinite-order class;
* it vanishes on all but finitely many finite-order classes.
For many years this was known only for groups whose Baum-Connes assembly
map behaves well. The theorem proved in this document removes that
hypothesis: it holds for every discrete group. The proof uses four ideas.
* Analytic correction: replace a general l^1 idempotent by a similar
one with exponential decay, without touching the trace.
* Cyclic encoding: rewrite each trace coefficient as a weighted sum
over cyclic tuples of group elements, with weights coming from
products of matrix coefficients.
* Differential forms: package that weighted sum as an average of
curvature Pfaffians over rotation-invariant one-forms.
* Multiscale smoothing: build those one-forms with a free parameter
(the time scale) so that their curvature shrinks with the simplex
dimension. The sum over tuples then converges to zero in the limit.
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1. GROUPS: THE ALPHABET OF SYMMETRY
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1.1 DEFINITION.
A group is a set G with
(i) a multiplication G x G -> G, (g, h) |-> gh;
(ii) an identity element e satisfying eg = ge = g for every g;
(iii) for every g, an inverse g^{-1} with gg^{-1} = g^{-1}g = e;
(iv) associativity, (gh)k = g(hk).
A group is discrete if we put no topology on G. All groups in this
document are discrete unless stated otherwise.
"Discrete" is important for two reasons. It means there is no
guarantee that infinite sums over G converge; that is exactly why the
l^1 norm appears. And it means that the only structure we can exploit
comes from the group law itself, not from any smooth or topological
data.
1.2 EXAMPLES TO KEEP IN MIND.
* Z, the integers, under addition. Identity 0, inverse -g.
* Z^n, n-fold product of Z, under addition.
* Z/nZ, the cyclic group of order n, under addition modulo n.
* F_2, the free group on two generators a, b. Its elements are
reduced words like a, b^{-1}, ab, a b a^{-1}. "Reduced" means
no adjacent generator and its inverse.
* S_3, the permutations of {1, 2, 3}. Non-commutative:
(12)(23) = (123), but (23)(12) = (132).
* Any concrete matrix group, e.g. SL_2(Z), if you want a rich example.
1.3 CONJUGACY CLASSES.
Conjugation is the operation g -> x g x^{-1}. Two elements are
conjugate if some x sends one to the other:
g ~ h iff h = x g x^{-1} for some x in G.
"~" is an equivalence relation. The conjugacy class of g, written [g],
is the set of all conjugates of g. The set of all conjugacy classes is
C(G); the finite-order classes form the subset C_fin(G). "Finite-order"
refers to element order, not to the size of the class.
The centralizer of g is
C_G(g) = { x in G : x g x^{-1} = g }.
For finite G, conjugation gives a bijection between [g] and the set of
left cosets G / C_G(g), hence the count |[g]| = |G|/|C_G(g)|. For
infinite G, the same bijection holds as a statement about sets.
The order of g is the least m >= 1 with g^m = e, or infinity if no such
m exists. An element (or class) is finite-order if its order is finite.
Two facts used repeatedly below:
* an identity g h = k can be rearranged h = g^{-1}k;
* the map x |-> x g x^{-1} is a bijection [g] -> [h^{-1}gh] for
every h; conjugation is an action of G on itself.
1.4 GROWTH.
A finitely generated group G comes with a word metric: pick a finite
symmetric generating set S, and let |g|_S be the length of the shortest
word in S that spells g. Then d(x, y) = |x^{-1} y|_S is left-invariant.
The ball B(x, s) = {y : d(x, y) <= s} of radius s grows at most like
exp(c s) for some constant c depending only on S: each step picks one
of |S| generators, so B(e, s) has at most (|S| + 1)^s elements. A
useful consequence:
|B(x, s)| <= exp(c_d s) for every s >= 1. (1.4.1)
Throughout the document, "c_d" and "gamma" denote small positive
constants that depend only on the word metric. They will absorb every
crude counting bound.
Any discrete group embeds in a finitely generated one in a way that
preserves the trace properties needed below (this is [BCM2004, Lemma
1.5]); we may therefore assume G is finitely generated whenever it
simplifies notation. This is not a hypothesis in the main theorem; the
reduction is automatic.
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2. GROUP RINGS: POLYNOMIALS THAT REMEMBER THE GROUP
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2.1 INFORMAL PICTURE.
A group ring is "the ring of polynomials whose coefficients are complex
numbers and whose variables are elements of G". If G = Z, this looks
like
a = sum over k in Z of a_k t^k, a_k in C,
where t is a placeholder for the generator 1 of Z. For a general group,
replace "t^k" by "the element k" and remember that multiplication is
the group product (not commutative in general). Two such forms combine
by the usual rules:
(a + b)_g = a_g + b_g,
(a * b)(g) = sum over hk = g of a_h b_k.
The second formula is convolution. It says: the coefficient of g in
the product is the accumulated coefficient of every way of splitting g
into a product h k.
2.2 THE ALGEBRAIC VERSION.
The complex group ring C[G] consists of all finite formal combinations
a = sum over g in G of a(g) g, a(g) in C,
with only finitely many a(g) nonzero, plus the addition and convolution
above. It is a complex algebra with unit 1 = e.
The l^1 completion l^1(G) consists of all functions a: G -> C with
||a||_1 = sum over g in G |a(g)| < infinity.
It contains C[G] as a dense subspace and is complete for ||·||_1, so it
is a Banach algebra: an algebra complete in a norm that is
submultiplicative, ||a * b||_1 <= ||a||_1 ||b||_1. Submultiplicativity
is immediate from the definitions:
||a * b||_1 = sum_g |sum_{hk=g} a(h) b(k)|
<= sum_g sum_{hk=g} |a(h)| |b(k)|
= sum_h sum_k |a(h)| |b(k)|
= ||a||_1 ||b||_1.
Only in the l^1 completion can we sum an infinite series of products
and be sure the sum makes sense in the algebra. That is exactly why
the l^1 form of the Bass conjecture is stronger than the pure algebra
version over C[G].
For q >= 1, write M_q(l^1(G)) for q x q matrices with entries in
l^1(G). It is a Banach algebra with the entrywise norm
||A||_1 = sum over i, j, g |A_{ij}(g)|.
Again ||AB||_1 <= ||A||_1 ||B||_1. The identity matrix has norm q, not
one; that appears only in constant factors of estimates and does not
affect anything qualitative.
2.3 CONCRETE EXAMPLES.
* G = Z. An element a = sum_k a_k t^k with sum_k |a_k| < infinity
is a summable function on Z, and the product is standard
convolution. This algebra is isomorphic to the disc algebra of
continuous functions on the unit circle via a |-> sum_k a_k z^k,
and a is invertible in l^1(Z) if and only if that transform has no
zero on |z| = 1 (Wiener's lemma).
* G = Z/2Z = {e, g} with g^2 = e. Then l^1(G) has two basis elements
e and g and is isomorphic, as an algebra, to C x C via
alpha e + beta g |-> (alpha + beta, alpha - beta).
So l^1(Z/2Z) splits as a product of two copies of C.
* G = F_n, the free group. l^1(F_n) is infinite-dimensional. The
finite-support part is the free algebra on n non-commuting
generators. The series 1 + a + a^2 + ... does not converge in l^1,
but sum_k rho^k a^k does for |rho| < 1.
* G a finite group like S_3. Then l^1(S_3) = C[S_3] is finite-
dimensional, and the norm is the usual coordinate norm on the
finite basis.
2.4 WHERE THE l^1 HYPOTHESIS BITES.
Every element of l^1(G) is a norm limit of finite-support elements, but
the limit is not finite-support in general. Two consequences drive
everything later:
* a matrix over C[G] with finite support stays finite-support after
any algebraic operation, whereas a matrix over l^1(G) with
exponential decay has a "tail" that must be tracked through every
step;
* a projective module over l^1(G) is generated by an idempotent
matrix; even when the idempotent has infinite support, we are
allowed to pair its coefficients against any bounded weight, so the
sums defining the trace converge absolutely.
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3. THE l^1 NORM: MEASURING SIZE
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3.1 THREE ESSENTIAL PROPERTIES.
(a) Absolute summability: sum_g |a(g)| < infinity. So every sum over
G converges absolutely and may be rearranged.
(b) Submultiplicativity: ||a * b||_1 <= ||a||_1 ||b||_1. This makes
l^1(G) a Banach algebra, and it makes the geometric series
(1 - b)^{-1} = sum_k b^k converge whenever ||b||_1 < 1.
(c) Exponential refinement. For a finitely generated G, put
||a||_c = sum over g |a(g)| exp(c |g|).
Since |gh| <= |g| + |h|, this weighted norm too satisfies
||a * b||_c <= ||a||_c ||b||_c. It is a genuine strengthening of
the l^1 norm, and it becomes usable after the reduction of
section 10.
3.2 WHY SUBMULTIPLICATIVITY LETS US INVERT.
If b lies in any Banach algebra with ||b||_1 < 1, then
(1 - b)^{-1} = 1 + b + b^2 + b^3 + ...
converges absolutely in the norm, and the sum really is the inverse.
This is more useful than it sounds: to invert a general element a it
is enough to write a = 1 - b with ||b||_1 < 1. Analytic correction
(section 10) is exactly a procedure that puts a given idempotent into
this form, at the cost of passing to a similar idempotent.
3.3 CONCRETE NUMERICAL EXAMPLE.
Take G = Z with generator t, and let a = 1 - t/2. Then
||a||_1 = 1 + 1/2 = 3/2 > 1,
so the trick of section 3.2 does not apply to a directly. But a has
the form 1 - b with b = t/2, and ||b||_1 = 1/2 < 1, so
(1 - b)^{-1} = 1 + b + b^2 + b^3 + ... = sum_{k >= 0} t^k / 2^k,
which has l^1 norm sum_k (1/2)^k = 2 < infinity. So a IS invertible in
l^1(Z) despite ||a||_1 > 1. The lesson: to invert 1 - b we need
||b||_1 < 1, not ||1 - b||_1 < 1.
Sanity check via the disc-algebra model. The element a has transform
1 - z/2. Its unique zero is z = 2, outside the unit circle, so by
Wiener's lemma a is invertible. Consistent.
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4. IDEMPOTENTS: PROJECTORS THAT NEVER DIE
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4.1 DEFINITION AND EXAMPLES.
An idempotent is an element p with p^2 = p. Examples:
* 0 and 1 in any ring.
* A diagonal matrix diag(1, 1, 0, ..., 0) in M_q(C).
* If G contains an involution g (g^2 = e), then (1/2)(e + g) is
idempotent in l^1(G). The coefficients are finite summable.
4.2 THE MODULE PICTURE.
Given an idempotent p in M_q(A) over a ring A, the set p A^q = {p v :
v in A^q} is a projective right A-module. Conversely, every finitely
generated projective A-module arises this way for some q and some
idempotent p, because projectives are direct summands of free modules,
and direct summands of A^q are exactly images of idempotents in
M_q(A). This is the algebraic version of the Serre-Swan correspondence.
Four facts about idempotents drive everything later.
* Two idempotents p, p' in M_q(A) give isomorphic modules if and only
if there are matrices a, b with p = ab and p' = ba. This is the
algebraic analogue of "conjugate projections".
* Conjugation by an invertible u changes p to u p u^{-1} without
changing the isomorphism class of the module.
* Over l^1(G), given an idempotent p and a matrix q close to p, there
is an invertible u close to the identity with u q u^{-1} idempotent.
This is the analytic correction of section 10.
* The trace of an idempotent matrix over l^1(G),
trace_C(p) = sum_{g in C} sum_{i=1}^q p_{ii}(g),
is invariant under conjugation by invertible matrices over l^1(G)
and additive under direct sum.
4.3 EXAMPLE: FINITE GROUP Z/2Z.
Let G = Z/2Z = {e, g}, and let p = (1/2)(e + g). Then
p(e) = 1/2, p(g) = 1/2,
trace_[e](p) = 1/2,
trace_[g](p) = 1/2.
Since the group has only two conjugacy classes, finite support is
automatic and the conjecture is trivial. Every finite group is likewise
trivial: the set of conjugacy classes is already finite. The content
is entirely in infinite groups.
4.4 EXAMPLE: INFINITE GROUP AND WHERE L^1 HAS BITE.
Take G = Z and p = 1 (the algebra unit). Then p(g) = 0 for g != 0 and
p(0) = 1, so trace_[0](p) = 1 and trace_C(p) = 0 for every other C.
This is finite-support with only the identity surviving.
More interesting examples over l^1(Z) force an infinite-support
idempotent. But by Wiener's lemma, the only idempotents in l^1(Z) are
the trivial ones (any {0,1}-valued continuous function on the circle
with summable Fourier coefficients is constant). So l^1(Z) gives no
nontrivial test cases, and neither do Z^n or other abelian groups:
their l^1 idempotents are all finite-support. Any genuine challenge
for the conjecture must live in a non-amenable group, where the
idempotents are genuinely infinite-support and traces can a priori
have infinite non-zero parts.
This is the historical difficulty of the conjecture: over amenable
groups the Baum-Connes assembly map decides the trace, but over
non-amenable groups the assembly hypothesis can fail and a different
argument is needed.
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5. THE HATTORI-STALLINGS TRACE
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5.1 WHAT WE WANT TO EXTRACT.
Let p be an idempotent q x q matrix over l^1(G). The ordinary matrix
trace sum_{i=1}^q p_{ii} is an element of l^1(G), but it depends on the
choice of basis of the module, so it is not a well-defined invariant of
the projective module p l^1(G)^q. The correct invariant remembers the
group and the conjugacy structure, as follows.
For a conjugacy class C of G, define the coefficient
trace_C(p) = sum_{g in C} sum_{i=1}^q p_{ii}(g).
The sum over g in C is absolutely convergent because p_{ii} is in
l^1(G), and the q x q sum is finite. So trace_C(p) is a well-defined
complex number. It is the "coefficient of C" in an element of
l^1(C(G)) -- the l^1 space of functions on conjugacy classes.
5.2 WHY IT IS A TRACE.
The sum defining trace_C is invariant under conjugation by any
invertible matrix u over l^1(G): computing
trace_C(u p u^{-1}) = sum_{g in C} sum_i (u p u^{-1})_{ii}(g)
and using the algebraic cyclicity of the ordinary matrix trace (which
holds entrywise) gives trace_C(u p u^{-1}) = trace_C(p). In particular,
if p and p' are similar idempotents, their traces agree on every
conjugacy class.
Additivity: if p and p' are idempotents of sizes q and q', the block-
diagonal matrix p (+) p' is idempotent, and trace_C(p (+) p') =
trace_C(p) + trace_C(p'). So the trace is additive over direct sums.
These two properties say: the assignment p |-> (trace_C(p))_C
descends to a well-defined invariant of the projective module p A^q,
and it is additive under direct sum. That is exactly what a trace on
K_0 has to be.
5.3 A CONCRETE COMPUTATION.
For G = Z/2Z and p = (1/2)(e + g) as in section 4.3, the trace
trace_[e](p) = p(e) = 1/2,
trace_[g](p) = p(g) = 1/2.
Both are complex numbers, both nonzero, both indexed by finite-order
classes. For any slightly different conjugate p' = u p u^{-1} of p,
the SAME values appear because we know trace is invariant under
conjugation. So the two numbers (1/2, 1/2) are an invariant of the
module p l^1(G).
For G = Z and p = 1, the idempotent corresponding to the rank-1 free
module, the trace is (1, 0, 0, ...): just the identity class
contributes. In fact every idempotent in M_q(l^1(Z)) has the form
(projection to a rank-r free summand) with r a non-negative integer,
and trace_C = 0 for C != [0], trace_[0] = r.
5.4 CONTINUITY.
If p_k -> p in the l^1 norm, then trace_C(p_k) -> trace_C(p) for every
C, since the sums defining trace_C are absolutely convergent and the
norm controls the absolute sum. So the trace map is continuous from
M_q(l^1(G)) to l^1(C(G)). This is why the l^1 completion is the right
home: it has the analytic properties needed for the constructions of
sections 10-16, while still having enough information to see every
coefficient.
This continuity is used heavily below, because all the estimates
produce bounds on the trace by passing to limits of finite
approximations.
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6. K_0 AND THE TRACE MAP
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6.1 PROJECTIVE MODULES AS THE INPUT.
Say you want to compare two finitely generated projective modules P
and P' over a ring A. The cleanest algebraic invariant is the
Grothendieck group K_0(A). It is built from equivalence classes [P] of
finitely generated projective modules, under two rules:
* [P (+) P'] = [P] + [P'], and
* [P] = [P'] whenever P and P' are isomorphic.
Then K_0(A) is the free abelian group on equivalence classes modulo
the direct-sum relation; formally, it is the abelian group generated
by symbols [P], with [P (+) P'] = [P] + [P'] as the only relation.
Equivalently, using the idempotent picture of section 4, every class
in K_0(A) is a difference [p] - [p'] of idempotents, with the rules
[p (+) p'] = [p] + [p'] and [p] = [p'] whenever p and p' are
similar.
6.2 THE TRACE ON K_0.
Return to A = l^1(G). For a fixed conjugacy class C, the trace
trace_C is a homomorphism of abelian groups from the collection of
idempotents to C, and it is additive under direct sum. Hence it
extends uniquely to a group homomorphism
trace_C: K_0(l^1(G)) -> C.
Collecting all classes, we get
HS^1 : K_0(l^1(G)) -> l^1(C(G)),
HS^1([p]) = (trace_C(p))_C.
That is the Hattori-Stallings trace. It is a well-defined map.
The image l^1(C(G)) is the Banach space of absolutely summable
functions on conjugacy classes, |(a_C)_C|_1 = sum_C |a_C|. The map
HS^1 is continuous for the obvious topologies and compatible with
direct sum.
There is also a purely algebraic version with C(G) as target and C[G]
as source, called HS_alg. The l^1 version contains more information
because the target allows infinite, summable support. That is exactly
the extra content of the l^1-Bass conjecture.
6.3 A CHECK THAT MATCHES EXPECTATION.
For G = Z, the classes [e] and [g] for g != e are singletons. Then
K_0(l^1(Z)) = Z, generated by [1] (the rank-1 free module), and
HS^1(n[1]) = n delta_[0],
the function that is n on [0] and 0 elsewhere. That is finite support
in the classes, on the finite-order class [0]. It agrees with what the
conjecture predicts.
For a general discrete group G, we have no such clean description of
K_0. The l^1-Bass conjecture is a strong statement about the image of
HS^1 for the entire Gi (not just its amenable part), where the structure
of K_0 is otherwise unknown.
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7. THE BASS CONJECTURE, PAST AND PRESENT
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7.1 THE ORIGINAL (ALGEBRAIC) VERSION.
Bass's original conjecture, published in 1976, was about the integral
group ring Z[G]. In that setting there is no l^1 completion and the
trace coefficients are computed from finite-support matrices. The
conjecture asserts:
Let e be an idempotent matrix over Z[G]. Then trace_C(e) = 0 for
every conjugacy class C other than the class of the identity.
This would identify the K_0 of Z[G] with the group cohomology of the
identity, so it is a strong statement. It is notoriously hard: it is
known for torsion-free groups in only a handful of cases, and it
remains open for groups as simple as F_2 x F_2.
7.2 THE COMPLEX VERSION.
Over C, Bass's conjecture (also called the complex Bass conjecture)
weakens the conclusion:
Let e be an idempotent matrix over C[G]. Then trace_C(e) = 0 for
every infinite-order conjugacy class C, and for all but finitely
many classes altogether.
The relaxation to complex coefficients permits the trace to survive on
finite-order classes, and the "all but finitely many" clause is a
finite-support assertion. The fact that finite-order classes are
allowed is essential: for G = Z/2Z, both classes survive and both are
finite-order.
7.3 THE l^1 VERSION (BCM 2004).
Over the completion l^1(G), the natural question becomes:
Let e be an idempotent matrix over l^1(G). Then trace_C(e) = 0 for
every infinite-order class C, and trace_C(e) = 0 for all but
finitely many classes.
This is Conjecture 2.2 of Berrick-Chatterji-Mislin 2004 ("BCM2004").
Because l^1(G) contains more idempotents than C[G], the l^1 conjecture
formally implies the complex one. BCM2004 proved it for any countable
group whose Bost assembly map is rationally surjective in degree 0.
That covers every amenable group (via Lafforgue 2002), and also many
groups that are not amenable, such as groups with property (T) plus a
good assembly behaviour.
The "finite support" part is not automatic. Absolute summability of
the trace just says sum_C |trace_C(e)| < infinity; it does not prevent
infinitely many nonzero entries of decreasing sizes. The BCM2004
formulation therefore insists on an actual finite set of surviving
classes, not just a summable tail.
7.4 OUR TARGET.
The conjecture we prove in this document is precisely the BCM2004
statement, for every discrete group. That removes the Baum-Connes
hypothesis (the "assembly map surjective" assumption). For groups
where the assembly hypothesis was needed, this is new. For groups
covered by BCM2004, we recover the result by a different route.
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8. THE MAIN THEOREM
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8.1 STATEMENT.
Theorem (l^1-Bass, every discrete group). Let G be any discrete group,
let q >= 1, and let e be a q x q idempotent matrix over l^1(G). Then
there is a finite set F of finite-order conjugacy classes of G such
that trace_C(e) = 0 for every C not in F. Equivalently, the image
HS^1(K_0(l^1(G))) lies in the algebraic direct sum over the finite-
order classes:
HS^1(K_0(l^1(G))) subset of (+)_C in C_fin(G) C.
8.2 WHAT "COMPLETE" MEANS HERE.
Both parts of the conclusion are affirmative:
(i) Every infinite-order class is dead: trace_C(e) = 0.
(ii) Every finite-order class is dead except possibly finitely many.
Both parts are non-trivial. Part (i) says there is no leakage into
infinite-order classes at all. Part (ii) says the surviving finite-
order classes are compactly supported in C_fin(G), not just summable.
For finite groups, C_fin(G) is finite and part (ii) is automatic.
For infinite groups with infinitely many finite-order classes -- for
example, a free product of two Z/2Z factors, where every nontrivial
element is conjugate to either of the factors' involutions or a
product -- part (ii) has real content: the theorem says the trace
cannot survive on unboundedly many classes even though there are
infinitely many.
8.3 WHAT DEPENDS ON WHAT.
The proof is organized to separate three influences:
* the analytic local structure of e (the reduction to exponential
decay in section 10), which is a purely Banach-algebraic step;
* the combinatorial structure of cyclic tuples of group elements
(section 11), which turns the trace into a weighted sum over
cyclic data;
* the geometric analysis of the resulting differential forms
(sections 12-15), which uses multiscale smoothing and integral
operator estimates.
The three parts are independent and are combined only at the end.
That structure is what makes the argument readable: you can follow
each piece in isolation and see how it plugs in.
8.4 WHAT IS NOT ASSUMED.
The theorem does not assume that G is amenable, torsion-free,
finitely generated, or of finite asymptotic dimension. It does not
assume Baum-Connes. It does not assume any of the analytic assembly
hypotheses of BCM2004. The single hypothesis is "G is a discrete
group".
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9. THE PROOF STRATEGY IN ONE PAGE
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The theorem has five steps. We state them here so the geography is
clear before we dive into the technical sections.
STEP 1 (section 10). REDUCE TO EXPONENTIAL DECAY.
Given an arbitrary idempotent e in M_q(l^1(G)), replace it by a similar
idempotent p lying inside M_q(l^1(Gamma)) for a finitely generated
subgroup Gamma, with exponential decay: sum_h |p(h)| exp(c|h|) <
infinity for some c > 0. Conjugation preserves every trace coefficient
by section 5. So it is enough to prove the theorem for such p.
STEP 2 (section 11). WRITE THE TRACE AS A TUPLE SUM.
Fix a group element g. The trace coefficient trace_[g](p) can be
written as an absolutely convergent weighted sum over cyclic tuples h =
(h_0, ..., h_k) of group elements whose closing edge is twisted by g.
Each tuple carries a weight v(h) = trace of an ordered product of
coefficients of p. Idempotency p^2 = p makes these weights compatible
with deleting vertices; cyclicity gives an additional rotation
symmetry. These structural facts are the engine of the whole proof.
STEP 3 (section 12). TURN THE FINITE-DIMENSIONAL SUM INTO A FORM.
On the space of ordered times 0 <= t_0 <= ... <= t_k <= t_0 + 1 one
puts a family of smooth one-forms (the "connection" forms), invariant
under simultaneous time translation and compatible with face deletion.
Stokes' theorem on each simplex turns the weighted trace sum into an
average of curvature Pfaffians. The (important) consequence is that the
trace coefficient equals an average of a scalar (the Pfaffian) against
the tuple weights.
STEP 4 (sections 13, 14). BUILD A GOOD CONNECTION VIA SMOOTHING.
The connection is built from the following picture. A "tuple" is a
step function on the circle, with values in G and jump points at the
times t_i. Average that step function over a sequence of time scales
delta_1 < delta_2 < ... < delta_b. Each average is a probability
distribution on G. Summarize it smoothly on an auxiliary label set
("channel"), which is what sections 13 does. Then use a one-form kernel
in time to convert those summarized distributions into a connection
with an explicit positive contraction B(V). The alternating sum over
stages telescopes thanks to the channel design (section 14).
STEP 5 (sections 15, 16). BOUND THE CURVATURE AND TAKE n -> infinity.
The resulting connection's curvature Pfaffian has three kinds of terms,
corresponding to three time scales:
* at the smallest scales, sparseness in time bounds the Pfaffian;
* at intermediate scales, the short time kernel produces a
Hilbert-Schmidt bound;
* at the largest scale, the residual "equality" kernel has singular
values of size 1/nu.
Combining these bounds gives a curvature estimate of size
(CS^{3/2} R^{-1/10})^n times an exponential factor in the tuple
lengths, where S = b_* + 2 is the number of stages. For large enough
R, this decays exponentially in n. Since the tuple sum has at most
M_p^{n+1} mass (M_p from the exponential moment of p), the sum over
n-tuples decays like (Q_R M_p)^n. Choosing R so that Q_R M_p < 1
forces trace_[g](p) = 0 for infinite-order classes.
The surviving finite-order classes are those whose conjugacy classes
avoid a fixed ball of radius 4r around the identity. There are only
finitely many such classes. This gives finite support in C_fin(G), and
section 16 assembles the result.
That is the plan.
The language is chosen to keep track of exactly two things throughout:
* a "scale" parameter (either the tuple dimension n or the simplex
time radius), and
* a "spatial" parameter R choosing the base radius.
Every estimate is of the form "polynomial in R^n, exponential in n,
absorbed into the free choice of the outer parameter".
================================================================================
10. STEP 1: REDUCING TO EXPONENTIAL DECAY
================================================================================
10.1 WHAT WE WANT TO REDUCE TO.
Let e in M_q(l^1(G)) be idempotent (so e^2 = e). We want to find:
* a finitely generated subgroup Gamma of G;
* a word length | | on Gamma;
* a positive number c > 0;
* a matrix p in M_q(l^1(Gamma)) with p^2 = p;
such that
* p has exponential decay: sum over h in Gamma of |p(h)|_1 exp(c|h|)
< infinity, where |p(h)|_1 = sum_{i, j} |p_{ij}(h)| is the
entrywise l^1 norm of the matrix coefficient;
* p is similar to e in the larger algebra M_q(l^1(G)).
"Similar" means there is an invertible matrix u in M_q(l^1(G)) with
p = u e u^{-1}. By section 5.2, this implies trace_C(p) = trace_C(e)
for every conjugacy class C meeting Gamma. For classes C not meeting
Gamma, trace_C(p) = 0 because p lives in Gamma. For classes C in G
that intersect Gamma, trace_C(e) = sum over classes D of Gamma with
D subset of C of trace_D(p): a finite-support reduction from a general
group to a finitely generated subgroup.
10.2 THE KEY: APPROXIMATE BY FINITE SUPPORT AND CORRECT.
Recall that C[G] is dense in l^1(G). So choose a finite-support
matrix a in M_q(C[G]) close to e. Close enough that the following
correction formula works:
b = 4(a - a^2)
is small, say ||b||_1 < 1. (We should also choose a so that this b is
small; that is achievable because a is close to e = e^2 and the map
a |-> 4(a - a^2) is continuous at e.) Then define
Q = (1 - b)^{-1/2} = sum over k >= 0 of (binom(2k, k) / 4^k) b^k,
p = 1/2 I + (a - 1/2 I) Q.
The scalar power series sum_k binom(2k,k) x^k / 4^k is the Taylor
expansion of (1 - x)^{-1/2}, valid for |x| < 1, so Q^2 (1 - b) = I
in the Banach algebra M_q(l^1(G)) because ||b||_1 < 1.
Idea: Why does p^2 = p?
We compute (p - I/2) = (a - I/2) Q and estimate
(p - I/2)^2 = Q (a - I/2)^2 Q.
Now (a - I/2)^2 = a^2 - a + I/4 = (I - 4(a - a^2))/4 = (1 - b)/4.
So
(p - I/2)^2 = Q (1 - b) Q / 4 = I / 4,
because Q^2 (1 - b) = I and Q commutes with itself. Hence
p^2 - p + I/4 = I/4, i.e. p^2 = p.
The argument does not use exponentiation of operators or any other
analysis; only the power series for (1 - b)^{-1/2}, which converges
because ||b||_1 < 1.
10.3 WHY p IS CLOSE TO e.
As a -> e, we have b -> 0 and Q = (1 - b)^{-1/2} -> I. Then
p = 1/2 I + (a - 1/2 I) Q -> 1/2 I + (e - 1/2 I) = e. So p is a norm
limit of the idempotents coming from a. Choose a sufficiently close to
e and p is close to e.
10.4 WHY p IS SIMILAR TO e.
Set u = p e + (1 - p)(1 - e). A direct computation (using pe = pu,
which follows from p being built from powers of a and hence commuting
with a) shows:
u e = p u.
Also
u - I = (p - e)(2e - I).
The second factor 2e - I is bounded (it has ||2e - I||_1 bounded by
constants depending only on e via |e|, ||e||_1), so for p sufficiently
close to e, the product (p - e)(2e - I) has norm < 1. Then u is
invertible (by the Neumann series) and
p = u e u^{-1}.
10.5 WHY p HAS EXPONENTIAL DECAY.
The matrix a has finite support, so it lives in M_q(C[F]) for a finite
set F. Let Gamma be the subgroup of G generated by the support of a
(which is finite because a is finite-support). Then Gamma is finitely
generated, and a in M_q(C[Gamma]).
Fixing a finite symmetric generating set of Gamma and its word length
| |, consider the weighted norm
||v||_c = sum_{h in Gamma} |v(h)|_1 exp(c|h|).
This norm is submultiplicative because |xy| <= |x| + |y| for the word
length. Since b has finite support, ||b||_c -> ||b||_1 as c -> 0. We
already have ||b||_1 < 1, so by continuity there is c > 0 with
||b||_c < 1. In that case the power series Q = (1 - b)^{-1/2}
converges in || ||_c, and hence Q, p, and (p - e)(2e - I) all have
finite || ||_c norm. So
sum_h |p(h)|_1 exp(c|h|) < infinity,
which is the exponential decay property.
10.6 PUTTING IT TOGETHER AND CARRYING TO G.
Proposition. Let e in M_q(l^1(G)) be idempotent. Then there exist a
finitely generated subgroup Gamma of G, a word length | |, a constant
c > 0, and an idempotent p in M_q(l^1(Gamma)) with exponential decay
such that p is similar to e in M_q(l^1(G)).
Transferring the trace: for a conjugacy class C of G,
trace_C(e) = trace_C(p) = sum over classes D of Gamma with D subset C
of trace_D(p).
Since every element of a finite-order class in Gamma has the same order
in G, and since Gamma has finitely many classes at a bounded distance
from the identity (being finitely generated), any finiteness result
established in Gamma will transfer to G. Likewise, absence of infinite-
order D in Gamma gives absence of infinite-order C in G: if C is an
infinite-order class of G intersecting Gamma, its intersection is a
union of infinite-order classes D of Gamma, each with trace_D(p) = 0.
So from now on:
* we may and do assume G is finitely generated;
* we may and do assume e has exponential decay: sum_h |e(h)|_1
exp(c|h|) < infinity for some fixed c;
* the word metric on G is fixed and all constants c_d, gamma below
are determined by it.
10.7 BOUNDS ON BALLS.
Two constants will frequently appear.
* Ball growth: since G is finitely generated, |B(x, s)| <= exp(c_d s)
for some c_d, uniformly in x and s >= 1. This is (1.4.1).
* Moment: define
M_p = sum_{w in G} |p(w)|_1 (1 + |w|).
Exponential decay with c > 0 gives M_p < infinity, and M_p will
control the sum of |v(h)| prod_i l_i in section 11. It is fixed
throughout: a single constant depending only on p and the word
metric.
With these conventions in place, the whole rest of the proof works
only with exponentially decaying idempotents. Any constant with a
subscript R may depend on the spatial radius R introduced later; no
constant depends on a tuple, its length n, or the twist element g.
================================================================================
11. STEP 2: TURNING THE TRACE INTO A TUPLE SUM
================================================================================
11.1 THE BASIC LEMMA.
Fix g in G and write H = C_G(g) for the centralizer. Consider the set
T_k(g) = H-orbits in G^{k+1}
of orbits of the group H^k+1-action on tuples (h_0, ..., h_k), where H
acts by simultaneous left multiplication: x . (h_i) = (x h_i).
Equivalently, T_k(g) is the set of (k+1)-tuples considered up to
simultaneous translation.
For a tuple h = (h_0, ..., h_k), put the indices in Z/k+1 by setting
h_{i+k+1} = g h_i
for both indices and edges, and define
v_{k+1}(h) = tr(prod_{i=0}^k p(h_i^{-1} h_{i+1})),
l_i = 1 + d(h_{i-1}, h_i).
Here "tr" is the ordinary complex matrix trace in M_q(C); the product
is over i in the displayed order; and l_i is the length of the edge
from h_{i-1} to h_i.
Both expressions descend to T_k(g). Indeed, replacing h by x h with x in
H does not change h_i^{-1} h_{i+1} because x cancels, and does not
change l_i because d(x a, x b) = d(a, b) by left invariance of the word
metric.
The factorization "closing edge is g" is built into the definition
h_{i+k+1} = g h_i. At the closing edge (i = k), the product includes
h_k^{-1} h_{k+1} = h_k^{-1} g h_0, which is exactly the twisted step.
11.2 WEIGHT IDENTITY.
Lemma. For every k >= 0 and every g, the following hold.
(i) The weighted sum over all cyclic tuples equals the trace:
trace_[g](p) = sum over h in T_k(g) of v_{k+1}(h).
(ii) Twisted rotation: the map
(h_0, ..., h_k) |-> (h_1, ..., h_k, g h_0)
preserves v_{k+1}. (This is just a cyclic rotation of the matrix
factors inside the matrix trace.)
(iii) Compatibility with deletion: for k >= 1, summing v_{k+1}(h) over
all insertions of one extra entry into a fixed tuple in T_{k-1}(g)
gives v_k of that fixed tuple.
Idea of proof.
For (i), the trace defining trace_[g](p) is
sum of tr(p(w_0) p(w_1) ... p(w_k))
over all (k+1)-tuples (w_0, ..., w_k) with w_0 w_1 ... w_k in [g].
Every such tuple determines h uniquely: choose h_0 in [g] and then put
h_{i+1} = h_i w_i. Two choices of h_0 differ by an element of H, so the
correspondence is a bijection T_k(g) <-> {(w_0, ..., w_k) : product in
[g]}. Summing over T_k(g) recovers the trace as stated.
For (ii), the matrix trace is cyclic: tr(A_0 A_1 ... A_k) is invariant
under cyclic reordering of the A_i. Replacing h by (h_1, ..., h_k, g h_0)
permutes the matrices w_i cyclically and inserts the closing twist, so
the trace is unchanged.
For (iii), summing over the insertion of one entry x between h_{i-1} and
h_i amounts to summing over all ways of writing the edge h_{i-1}^{-1}
h_i as h_{i-1}^{-1} x . x^{-1} h_i and using the convolution identity
sum over x in G of p(ax) p(x^{-1} b) = p^2(ab) = p(ab)
for every a, b. Repeating for every insertion position gives the
deletion compatibility.
11.3 ABSOLUTE CONVERGENCE AND MOMENT BOUND.
Every product of matrix coefficients p(h_i^{-1} h_{i+1}) is dominated by
the product of |p(h_i^{-1} h_{i+1})|_1, and the sum over all tuples of
any fixed product of |p(.)|_1 is the total mass of p^{k+1} = p, which is
finite. So the sum in 11.2(i) is absolutely convergent.
Further, multiplying by prod_i l_i does not destroy convergence. Because
l_i = 1 + |h_{i-1}^{-1} h_i|_S, and p has exponential decay, the
moment
M_p = sum_w |p(w)|_1 (1 + |w|)
is finite, and the following crude bound holds:
sum over h in T_k(g) of |v_{k+1}(h)| prod_i l_i <= M_p^{k+1}.
(11.3)
Similarly, multiplying by any fixed polynomial in the edge lengths
does not affect convergence, because |v_{k+1}(h)| <= prod_i |p(w_i)|_1
and the polynomial factor can be absorbed into a slightly faster
exponential decay.
11.4 WHY THIS HELPS: EVERY TRACE COEFFICIENT IS A SUM OVER TUPLES.
To summarize: for each fixed g and every k, the trace coefficient
trace_[g](p) equals a sum of v_{k+1} over all tuples in T_k(g). The
tuples are symmetric under rotation and compatible with vertex deletion.
The final proof of the theorem will run by taking k (or a related
dimension n) to infinity and showing the right side converges to zero.
================================================================================
12. STEP 3: FROM TUPLE SUM TO CURVATURE AVERAGE
================================================================================
12.1 ORDERED TIMES ON THE SIMPLEX.
Given a tuple h in T_k(g), we want to attach a continuous "timeline" to
the tuple, so that the sum over tuples becomes an average of a scalar
over a simplex. Introduce the ordered time domain
Delta~_k = {(t_0, ..., t_k) : t_0 <= t_1 <= ... <= t_k <= t_0 + 1}.
The circle R/Z has length 1; time is periodic. Extend times by
t_{i+k+1} = t_i + 1. Define gaps
x_i = t_{i+1} - t_i, 0 <= x_i, sum_i x_i = 1.
Then gap x_i = 0 has a natural geometric meaning: it corresponds to
deleting vertex h_i from the tuple. The face "x_i = 0" of the simplex
is exactly the simplex of the shortened tuple.
The translation group R acts by adding the same constant to all t_i.
Its infinitesimal generator is V = sum_{i=0}^k d/dt_i. The quotient is
the standard simplex
Delta_k = {0 <= t_1 <= ... <= t_k <= 1},
with orientation dt_1 ^ ... ^ dt_k. Sometimes we identify Delta~_k with
its section t_0 = 0 in Delta_k.
12.2 DEFINITION: A COMPATIBLE CONNECTION FAMILY.
A "compatible connection family through dimension n" assigns, to each
tuple h in T_k(g) with 0 <= k <= n, a real 1-form alpha_h on Delta~_k,
subject to four properties:
(i) Translation invariant: alpha_h is invariant under simultaneous
translation of the t_i, and alpha_h(V) = 1. (Think of alpha_h as
a "gauge field" on the timeline; alpha_h(V) = 1 lets it be used
as a bookkeeping coordinate for time.)
(ii) Centralizer-invariant and rotation-equivariant: alpha_{x . h}
= alpha_h for x in H, and alpha_{rho(h)} = alpha_h under the
twisted rotation of 11.2(ii).
(iii) Face-compatible: the tangential restriction of alpha_h to the
face x_i = 0 equals alpha_{h'} for the tuple h' obtained by
deleting (h_i, t_i).
(iv) Smooth with polynomial bounds: for each fixed k and each fixed
construction parameter, alpha_h and its exterior derivative
d alpha_h have coefficients that are smooth up to every face (as
one-sided limits) and bounded by a polynomial in the edge lengths
l_i.
Two consequences of (i):
* alpha_h is essentially a 1-form on Delta_k: after fixing t_0 = 0,
it is invariant under translation, so it descends;
* if we set A_h = matrix of d alpha_h in the section t_0 = 0, then
A_h is an odd (antisymmetric) k x k matrix, and A_h is a form on
Delta_k.
12.3 FROM ONE-FORMS TO TRACE COEFFICIENTS.
The following is the key bridge. Reference: [cy:trace-identity] in the
original paper.
Proposition (trace as curvature average). Let n = 2m > 0 be even and
let alpha be a compatible connection family through dimension n. For
every tuple h in T_n(g), define A_h as above. Then
sum over h in T_n(g) of v_{n+1}(h) . E_{Delta_n} Pf(A_h)
= (-1)^m trace_[g](p). (12.3)
Here E_{Delta_n} is uniform probability on Delta_n, and Pf is the
Pfaffian: for a real skew-symmetric 2m x 2m matrix A,
Pf(A)^2 = det(A), and the scalar property Pf(A) = +-sqrt(det(A)) is
chosen continuously along skew matrices; concretely
Pf(A) dt_1 ^ ... ^ dt_{2m} = (1/m!) (sum_{i < j} A_{ij} dt_i ^
dt_j)^m.
The sum on the left is absolutely convergent.
Idea of the proof (3 short steps).
Step A: Two compatible families give the same result. Compare two
compatible families alpha and alpha'. Their difference is closed on
each fibre of Delta~_k -> Delta_k (a "basic" form). Put
eta = (alpha - alpha') ^ (sum_{j=0}^{m-1} (d alpha)^j ^
(d alpha')^{m-1-j}).
This is the Chern-Simons transgression form of a difference of
connections; it is basic degree 2m-1. Its exterior derivative is
d eta = (d alpha)^m - (d alpha')^m.
Integrating over Delta_n (Stokes) and multiplying by v_{n+1}(h), then
summing over tuples, the left side collapses to a signed sum of boundary
contributions with v_n(h') at the boundary tuples h'. This is exactly
the deletion compatibility of 11.2(iii), so we can identify the
contribution with
I = sum over h' in T_{n-1}(g) of v_n(h') int_{Delta_{n-1}} eta_{h'}.
Step B: The boundary sum is zero. On Delta_{n-1}, the twisted rotation
permutes the n gap coordinates (i.e. the n coordinates t_1, ..., t_n)
cyclically. Since n - 1 is odd (because n is even), the cyclic
permutation has orientation sign (-1)^{n-1} = -1. All the data (v_n,
eta, the face identifications) is invariant under rotation by
property (ii), so the integral gets an extra sign change:
I = -I,
hence I = 0. So sum_h v_{n+1}(h) int_{Delta_n} (d alpha_h)^m is
independent of the choice of compatible family.
Step C: Evaluate at a convenient family. The canonical family
alpha^can = sum_{i=0}^k x_i dt_i
(with x_i the gap coordinates) satisfies all four requirements, and its
curvature on Delta~_k is
d alpha^can = sum_i dt_{i+1} ^ dt_i.
On Delta_n with t_0 = 0, the only complete pairing of the n time
differentials uses adjacent pairs (1,2), (3,4), ..., (n-1,n), each with
coefficient -1. So the Pfaffian of the curvature matrix is (-1)^m.
Since the uniform probability on Delta_n has total coordinate volume
1/n!, the standard identity
(n! / m!) int_{Delta_n} (d alpha_h)^m = E_{Delta_n} Pf(A_h)
follows from the Pfaffian volume formula. Combining with the total
weight identity 11.2(i) gives (12.3).
12.4 WHAT THIS BUYS US.
The proposition reduces the whole proof of the l^1-Bass conjecture to a
single analytic problem: build compatible connection families, in
arbitrarily large even dimensions, whose curvature Pfaffian has small
average size. If we can construct families with
E_{Delta_n} |Pf(A_h)| <= Q_R^n prod_i l_i (12.4)
for a positive Q_R (depending on a spatial scale parameter R and not on
h, n, or g), then using (12.3) and (11.3):
|trace_[g](p)| <= sum_h |v_{n+1}(h)| E |Pf(A_h)|
<= Q_R^n sum_h |v_{n+1}(h)| prod_i l_i
<= M_p . (Q_R M_p)^n.
Taking n -> infinity (through even integers) shows trace_[g](p) = 0
whenever Q_R M_p < 1.
Sections 13-15 produce such families and prove (12.4). The next section
constructs the "channels" -- the smooth summaries of probability
distributions on G -- on which the connection family will be built.
================================================================================
13. STEP 4A: SPATIAL SUMMARY CHANNELS
================================================================================
13.1 THE PROBLEM: HOW TO "SEE" A PROBABILITY ON G SMOOTHLY.
We want to build the compatible connection of section 12 by averaging
the step function coming from a tuple (section 14). At any time u, that
step function is a Dirac mass delta_{f(u)} at some group element f(u).
We need to convert these Dirac masses into something SMALLER in
dimension -- a summary distribution on an auxiliary space -- while
preserving enough information about spatial concentration and while
having controlled derivatives.
A "channel" Y is a map
Y: {finitely supported probabilities on G} -> {subprobabilities on
an auxiliary set Lambda}
that is (a) left equivariant under G acting on both arguments, and
(b) smooth on each finite-dimensional slice. The auxiliary set Lambda
is called the label set. We want channels with:
* sachannel bounds big enough to detect spatial concentration;
* derivative (Fisher) bounds small enough for the analytic estimates
of chapters 14-15.
13.2 THE CONCENTRATION FUNCTIONAL.
Fix a spatial radius r >= 1 and a piecewise-linear taper
chi_r(s) = min(1, max(0, 2 - s/r)), s >= 0.
Then chi_r(s) equals 1 for s <= r, 0 for s >= 2r, and interpolates
linearly in between. Its Lipschitz constant is 1/r.
For a probability P supported on a finite subset of G, define the
score and concentration functionals
E_r(P, y) = sum_h P_h chi_r(d(h, y)),
q_r(P) = max_{y in G} E_r(P, y),
phi_r(P) = (q_r(P) - a_*)_+, a_* = 0.80.
Interpretation: think of dropping a "fuzzy ball" of radius 2r at each
sampling point y and asking how much of P it catches. E_r(P, y) is the
amount caught. q_r(P) is the maximum over placements. phi_r(P)
measures how much of that maximum exceeds a fixed threshold a_* = 0.80.
Facts from the taper:
* E_r(P, y) ranges in [0, 1];
* q_r(P) exists and is attained on a finite set (in fact within the
2r-neighborhood of supp P);
* 0 <= phi_r(P) <= 1 - a_* = 0.20.
13.3 THE HIGH-LEVEL CHANNEL LEMMA.
The following is Proposition ch:summary of the original paper, stated
as the exact bound we need.
Proposition (spatial summary channels). Fix r >= 1 and 0 < epsilon <=
10^-3. There exist channels A, L and a cross kernel kappa such that,
in either of two regimes:
* (early) the resulting prediction uses radius r' = 6r;
* (later) the resulting prediction uses radius r' = r;
the following bounds hold for every P and every y in G.
(i) selection:
sum_z A(P)_z (E_r(P, z) - a_*) >= phi_r(P) - 10 epsilon.
(ii) prediction upper:
pred(P, y) <= E_{r'}(P, y) + 10 epsilon,
where pred(P, y) = sum_omega kappa(y, omega) L(P)_omega is the
prediction of the channel L on input P at reader y.
(iii) prediction lower:
pred(P, y) >= E_r(P, y) - 10 epsilon whenever A(P)_y > 0.
Additionally, at any positive output coordinate the derivative of the
channel with respect to the input probability vanishes when the output
does, and the Fisher seminorm (section 13.6) satisfies
(iv) F_Y(P; h, h') <= C(1 + d(h, h')) epsilon^{-10} in the early
regime, or C(1 + d(h, h')) r^{10} epsilon^{-10} in the later
regime.
The constants C depend only on the word metric, and the numerical
constants 10, 0.80 are independent of the group.
13.4 IDEA: THE SELECTOR CHANNEL A.
The first channel A is a soft selector: it picks out the points z
where the score E_r(P, z) is large, and puts small weight on the rest.
Concretely, take smooth cutoffs
psi_A(s) = 0 for s <= 0.70, 1 for s >= 0.75,
psi_D(s) = 0 for s <= 0.58, 1 for s >= 0.62,
(the two are essentially "soft staircase" functions; the specific
thresholds come from choosing different phases for different channels).
Then define unnormalized weights
w^A_z = psi_A(E_r(P, z))^2 * exp(N (E_r(P, z) - 0.80)),
w^D_z = psi_D(E_r(P, z))^2 * exp(N (E_r(P, z) - 0.66)),
N = M_0 r epsilon^{-2},
for a constant M_0 fixed by the word metric (M_0 = 25 gamma + 4 is
enough). Then normalize:
A(P)_z = w^A_z / (1 + sum_u w^A_u),
D(P)_z = w^D_z / (1 + sum_u w^D_u).
The extra 1 in each denominator is the weight of an unused "null" label
in the auxiliary set. It ensures A and D can have total mass much less
than one when there is no strong concentration.
Why these formulas. The exp factor is a "softmax": if one z has
significantly higher score than all rivals, its weight dominates and A
concentrates almost entirely on that z. If many z have similar scores,
the mass spreads out. The cutoff psi_A ensures that centers with E_r(P,
z) well below 0.75 carry no mass (their cutoff is zero). Similarly for
D but with lower thresholds.
By the elementary fact that at most exp(gamma r) points have score
above 1/2 (from the ball-growth bound of 1.4.1), only polynomially
many centers can carry mass. The exact estimates are:
Lemma (selector estimates). For every P and 0 < epsilon <= 10^-3:
* selection: sum_z A(P)_z (E_r(P, z) - a_*) >= phi_r(P) - 3 epsilon;
* D is supported near a concentrated point: sum_z D(P)_z >= 1 - epsilon
and sum_z D(P)_z E_r(P, z) >= q_r(P) - 2 epsilon whenever
q_r(P) > 0.70;
* derivatives: the Fisher seminorm (13.6.1) of A + D is bounded by
C epsilon^{-2} (1 + d(h, h')).
These are the "channel derivative" bounds of part (iv) of the
proposition with the early-regime epsilon-dependence.
13.5 THE PREDICTION CHANNEL L: TWO VERSIONS.
The channel L is built from D. It gives, for each input P, a
subprobability on a richer label set, weighted by a fixed kernel kappa
that predicts scores at group points. Two regimes give two different
constructions.
13.5a EARLY VERSION (r' = 6r).
This is the simpler construction. Set
Lambda = G,
L(P)_z = D(P)_z E_r(P, z),
kappa(y, z) = 1 if d(y, z) <= 4r, 0 otherwise.
Then the prediction at reader y is
pred(P, y) = sum_z D(P)_z E_r(P, z) 1_{d(y, z) <= 4r}.
Interpretation: L carries the "score information" D has collected, from
its centers x. A reader at y reads off those scores if y is within 4r of
x. Because the taper chi_r has support inside radius 2r, a center x that
is within 4r of y will have all its contributing h also within 6r of y.
So we get pred(P, y) <= E_6r(P, y) -- the prediction upper bound with
r' = 6r.
When y is a selected center (A(P)_y > 0), D's mass is essentially
concentrated near y and pred(P, y) >= E_r(P, y) - 2 epsilon. This is the
prediction lower bound with r' = 6r.
Derivative bound: from (13.6.1) applied to L_z = D_z E_z, using
|dot(E_z)| <= d(h, h')/r from the taper, one gets the early-regime
bound epsilon^{-4} on the Fisher seminorm.
13.5b LATER VERSION (r' = r): SAMPLE STRINGS.
The early version enlarges the radius by a factor of 6, which
would cost exponentially many centers in the derivative bound if used
at every stage. To keep r' = r, we instead use a finite random sample
of small strings.
Idea. Fix a center z and consider the truncated probability P^z = P
restricted to the ball B(z, 10r) (with the leftover mass assigned to a
"null" symbol, so the total is one). The truncated score E_r(P^z, y)
equals the full E_r(P, y) whenever y is within 4r of z -- exactly the
case we care about. We then approximate E_r(P^z, .) uniformly on the
ball B(z, 22r) by an empirical average
H_y(w) = (1/b) sum_{i=1}^b chi_r(d(w_i, y)) (w_i the samples)
from a string w = (w_1, ..., w_b) drawn from P^z. The discrepancy
Delta(P; z, w) = max_y |H_y(w) - E_r(P^z, y)| can be made <= epsilon
with positive probability by Hoeffding, for b of order r epsilon^{-4}.
So SOME string w is epsilon-accurate.
To make the assignment of w to a smooth function of P, use a Gibbs
weighting on the finite set of all strings in a fixed alphabet:
pi_z(P)_w ~ exp(-N_1 S(P; z, w)),
where S is a smooth surrogate for Delta (using log-sum-exp to replace
max), and N_1 = M_2 r b epsilon^{-2} is a large penalty so that mostly
only the accurate strings get weight. Then the empirical average of
Delta over pi_z(P) is small (<= 5 epsilon), and the derivative of
pi_z(P)_w with respect to a coordinate of P is bounded by N_1 d(h,h')/r.
Now set
Lambda = {(z, w) : z in G, w in A_z^b},
L(P)_{z, w} = D(P)_z * pi_z(P)_w,
kappa(y, (z, w)) = H_y(w).
Prediction bounds:
* pred(P, y) <= E_r(P, y) + 5 epsilon: from the string bound;
* pred(P, y) >= E_r(P, y) - 6 epsilon when A(P)_y > 0: from D's
concentration and the truncation-exactness E_r(P^z, y) = E_r(P, y).
Derivative bound: F_L(P;h,h') <= C r epsilon^{-6}(1 + d(h,h')). This is
the later-regime estimate.
13.6 THE FISHER SEMINORM.
To state derivative bounds, one uses an information-theoretic seminorm.
For a channel Y, a probability P and a "transfer direction" dot P =
delta_h - delta_{h'}, put
F_Y(P; h, h') = (sum_{omega : Y(P)_omega > 0} |dot Y(P)_omega|^2 /
Y(P)_omega)^{1/2}. (13.6.1)
This is the square root of the Fisher information of the family
P |-> Y(P) in the direction delta_h - delta_{h'}. Two properties are
used:
* it is invariant under scaling, so all the estimates are dimensional;
* if a channel weight goes to zero, a smooth channel has its
derivative also go to zero, so no term blows up at zero.
The Fisher seminorm appears in the proof of sections 14-15 as follows.
The exterior derivative of a channel-valued form is controlled by
F_Y times the sum of |d u| in the direction of a coordinate of P. So
control on F_Y directly bounds the curvature Pfaffian later.
13.7 TWO DISJOINT CHANNELS AND THE SYMMETRIC KERNEL.
When combining A and L into a single object Y = A (+) L, one has to
keep track of pairings. Set
Y(P) = A(P) (+) L(P),
on the disjoint union Lambda = G (+) Lambda_L (even though both label
sets may be G). Define a symmetric kernel K on this union by:
K(y, omega) = K(omega, y) = (1/2) kappa(y, omega) for y in G,
omega in Lambda_L;
K = 0 within either channel.
Then the "diagonal" bilinear pairing of K with Y is
K(Y(P), Y(P)) = sum_y A(P)_y * pred(P, y).
This is exactly the "sum of selected score times prediction" that
appears in the trace formula of section 14. The combined field has
mass at most two, and the Fisher seminorm of the combined field is
sqrt(F_A^2 + F_L^2), controlled by the proposition.
13.8 WHAT THE CHANNEL DOES FOR US.
Pulling all this together:
* the A channel detects the concentration phi_r(P) up to error 10 eps;
* the L channel gives an upper bound E_{r'}(P, y) + 10 eps and a
lower bound E_r(P, y) - 10 eps at positions where A is supported;
* the derivatives are bounded by constants that lose at most a
power of r and a power of epsilon.
The next section combines these into the multiscale averaging
construction: one layer of A/L per time scale, telescoping the sum of
"concentration at scale j" minus "concentration at scale j+1" down to
a strictly positive final amount. This positivity is what turns a
one-form B into a normalized connection alpha = B / B(V) with
B(V) >= c_0 > 0.
================================================================================
14. STEP 4B: THE AVERAGING CONNECTION
================================================================================
14.1 PARAMETERS.
Fix a spatial radius R >= 1 (large; chosen at the end of the argument).
Choose:
* an integer J_* >= 1 with (3/2)^{J_*-1} > 200;
* an accuracy parameter 0 < epsilon_* <= 10^-3 (small);
* integers b_* >= J_* and a stage count J_* < b_* (see below).
Spatial radii:
r_j = 6^{min(j, J_*)-1} R for j >= 0,
r = 6^{J_*-1} R.
So r_1 = R; r_j grows by a factor of 6 for the first J_*-1 stages,
then stays at r = r_{J_*} for every later stage.
Accuracies: epsilon_j = epsilon_* / (1 + j)^2 for j >= 1. These sum
(over j) to a value bounded by (pi^2/6) epsilon_*, which we choose small
enough that 30 sum_{j>=1} epsilon_j <= 1/10.
Multiplier: a fixed constant C_* (larger than an explicit but
uninteresting bound from section 13) and
W = exp(C_* r).
Time scales: choose a base ladder
beta_1 = R,
beta_{j+1} = beta_j^{3/2},
and let b_* be the smallest integer with
b_* >= J_* and beta_{b_*} >= W^2 R^{100}.
Since log beta_j = (3/2)^{j-1} log R, this gives b_* = O(1 + log R).
For an even integer n > 10 beta_{b_*} (the "dimension") set
delta_j = beta_j / n for 1 <= j <= b_*.
Then delta_1 < delta_2 < ... < delta_{b_*}, and:
* sum_{i<=j} delta_i <= 2 delta_j for every j (by
beta_1 + ... + beta_{j-1} <= beta_j);
* 10 delta_{b_*} < 1 (by n > 10 beta_{b_*}).
These are the exact hypotheses needed for the averaging construction.
SAME PARAMETERS IN EVERY DIMENSION. The values of all the above
parameters are fixed once R and n are chosen. In particular, when we
later restrict a form to a face of a simplex (a lower-dimensional
simplex), we do NOT change the delta_j; the construction is consistent
with the face compatibility demanded in Definition 12.2.
14.2 THE STEP FUNCTIONS AND TIME AVERAGES.
Given a tuple h = (h_0, ..., h_n) in T_n(g) and times t_0 <= ... <= t_n,
define the step function
f(u) = h_i on [t_i, t_{i+1}), and f(u+1) = g f(u).
Moments become Dirac masses:
P_0(u) = delta_{f(u)}.
Choose a non-negative, even, C-infinity smoothing kernel mu supported
in [-1, 1] with integral 1. Set mu_j(s) = delta_j^{-1} mu(s / delta_j)
(scaling so that int mu_j = 1). Define successive averages:
P_j(u) = (mu_j * P_{j-1})(u - .) evaluated at u
= int_R mu_j(s) P_{j-1}(u - s) ds.
These are time-dependent probability distributions on G, and
P_j(u + 1) = g P_j(u) follows from the invariance of P_0. If we set
eta_j = mu_j * mu_{j-1} * ... * mu_1, then
P_j = eta_j * P_0,
supp(eta_j) subset [-2 delta_j, 2 delta_j],
||eta_j||_infinity <= C / delta_j.
The support condition uses the scale ladder; the sup-norm bound is a
routine convolution estimate.
14.3 CHANNEL FIELDS.
For 1 <= j < b_*, apply Proposition 13.3 to P_j(u) at parameters
(r_j, epsilon_j). Use the early version (13.5a) when j < J_*, and the
later version (13.5b) when j >= J_*. Denote the resulting channels by
A_j, L_j, with prediction kernel kappa_j. Write
Y_j(u) = A_j(P_j(u)) (+) L_j(P_j(u)) on the disjoint union of the
two label sets.
Define the symmetric kernel K_j by
K_j(y, omega) = (1/2) kappa_j(y, omega) for y in G, omega in Lambda_j,
K_j = 0 within A_j or L_j.
Then K_j(Y_j(u), Y_j(u)) = sum_y A_j(P_j(u))_y * pred_j(P_j(u), y).
The total mass of Y_j(u) is at most two, and |K_j| <= 1.
At the endpoints, set
Y_0(u) = P_0(u), K_0(h, h') = chi_R(d(h, h')),
Y_{b_*}(u) = D(P_{b_*}(u)), K_{b_*}(h, h') = 1_{h = h'}.
These fields have mass at most one. All fields obey Y_j(u+1) = g Y_j(u).
All kernels are invariant under simultaneous left translation.
14.4 THE LAG KERNELS.
For 0 <= j < b_* define the odd kernel
U_j(s) = sign(s) - 2 int_0^s rho_j(a) da with rho_j = mu_{j+1} *
mu_{j+1}.
Properties:
* |U_j| <= 1;
* U_j = 0 outside [-2 delta_{j+1}, 2 delta_{j+1}];
* in the sense of distributions:
U_j' = 2(delta_0 - rho_j).
The last identity is what turns later contractions into differences of
consecutive pairings, giving the telescoping.
At the last stage:
* if g has infinite order, set U_{b_*}(s) = sign(s) and rho_{b_*} = 0;
* if g has finite order, set U_{b_*}(s) = sign(s) 1_{|s| < 1} and
rho_{b_*} = (1/2)(delta_{-1} + delta_{+1}).
The identity U_{b_*}' = 2(delta_0 - rho_{b_*}) also holds at the last
stage.
14.5 THE FORMS B.
For each stage 0 <= j <= b_* define the one-form (in the tuple times
t_0, ..., t_n, at fixed u):
B_j = -(1/2) int_per du int_R dv U_j(v - u) K_j(dY_j(u), Y_j(v)).
The integral "int_per" is over one period of the periodic function
u -> ..., using a half-open period when integrating the raw impulse
measure of P_0. The form dY_j is the exterior derivative with respect
to the tuple times only (u and v are held fixed).
Set
B = sum_{j=0}^{b_*-1} B_j + W B_{b_*}.
The multiplier W = exp(C_* r) is present at the last stage because D's
possible mass spread over exp(gamma r) labels must be compensated for.
14.6 COMPATIBILITY.
Proposition. B is a compatible connection-vector family (Definition
12.2 with the adjustment that alpha := B / B(V) will be the connection).
Specifically:
* B has translation-invariant coefficients in the tuple-times, with
well-defined smooth extensions to every face of every simplex;
* B is centralizer-equivariant and rotation-equivariant;
* B(∂/∂t_i) and its exterior derivative have coefficients bounded by
C_R l_i;
* the contraction C_nor := B(V) on the section t_0 = 0 satisfies
C_nor >= c_0 = 1/10, uniformly in n, g, and the tuple.
Idea of proof. Regularity and face compatibility follow from the
explicit formulas for the smoothed distributions P_j, from the smoothness
of the channels A_j, L_j, and from the delta identity at the raw stage.
Coefficient bounds: differentiated through the definitions, the only
"spike" comes from the raw P_0 impulses and from the bounded-lag U_j;
sections 14.4.1-14.6.1 of the paper compute the constants. We separate
those routine bounds from the important positivity statement, which is
content of 14.7.
14.7 CONTRACTION AND POSITIVITY.
Define the periodic scalar functions
Phi_j = int_per phi_{r_j}(P_j(u)) du for 1 <= j <= b_*.
Lemma (23.7 in the paper). The contraction decomposes as:
* C_0 = B_0(V) satisfies C_0 >= 1/5 - Phi_1;
* for 1 <= j < b_*, C_j = B_j(V) satisfies
C_j >= Phi_j - Phi_{j+1} - 30 epsilon_j;
* C_{b_*} = B_{b_*}(V) satisfies W C_{b_*} >= Phi_{b_*} (in either
of the two cases, infinite-order or displaced finite-order g).
Summing these:
C_nor >= 1/5 - 30 sum_{j=1}^{b_*-1} epsilon_j >= 1/10.
Idea of C_j. The clever choice of the lag kernel U_j gives, in the
distributional identity U_j' = 2(delta_0 - rho_j), a direct comparison:
C_j = int_per [K_j(Y_j(u), Y_j(u))
- int_R K_j(Y_j(u), Y_j(u + s)) d rho_j(s)] du.
The second pairing can be rewritten (using that mu is even and
rho_j = mu_{j+1} * mu_{j+1}) as
int_per K_j((mu_{j+1} * Y_j)(u), (mu_{j+1} * Y_j)(u)) du.
So C_j = int_per [diagonal pairing] - [averaged pairing], which by the
channel estimates and the telescoping design of A/L is at least
Phi_j - Phi_{j+1} - 30 epsilon_j.
Idea of C_0. The same identity with rho_0 in place of rho_j is
C_0 = 1 - int_per K_0(P_1(u), P_1(u)) du,
because the raw stage diagonal pairing is K_0(P_0, P_0) = 1. Then
K_0(P_1, P_1) = sum_y (P_1)_y chi_R(P_1, y) <= q_R(P_1) <= a_* +
phi_R(P_1), and choose P_0 = P_1 in the section t_0 = 0 of the
simplex. This yields C_0 >= 1 - a_* - Phi_1 = 1/5 - Phi_1.
Idea of W C_{b_*}. In the infinite-order case, the second pairing
vanishes (rho_{b_*} = 0), so C_{b_*} = int_per sum_z D(P_{b_*}(u))_z^2.
The channel D's terminal mass on labels satisfies
sum_z D_z^2 >= (1/4) exp(-gamma r) (from its support cardinality). So
W C_{b_*} = exp(C_* r) C_{b_*} >= exp((C_* - gamma) r) int_per phi_r du
>= Phi_{b_*}, since C_* >= gamma and phi_r <= 1/5.
In the finite-order case with d(h, gh) > 4r, D is supported on a set of
groups of diameter <= 4r, so D cannot have overlap with gD or g^{-1}D
(they would force an element of word length < 4r). Either lag pairing
at s = +-1 is zero, and the same argument applies.
14.8 THE NORMALIZED CONNECTION.
Let
alpha = B / C_nor.
By 14.7, C_nor >= c_0 = 1/10 is strictly positive, and C_nor is a
periodic scalar function of the tuple times. Dividing by it preserves
face compatibility and rotation equivariance. Also alpha(V) = 1 by
construction. So alpha is a compatible connection family in the sense
of Definition 12.2, and Proposition 12.3 applies.
By 12.4, the whole proof of the l^1-Bass conjecture reduces to:
Estimate the average over Delta_n of |Pf(d alpha_h)|, for each fixed
tuple h, in terms of a base Q_R that can be made < 1 / M_p by taking
R sufficiently large.
That is the content of the next section.
================================================================================
15. STEP 5: CURVATURE ESTIMATES AND THE PFAFFIAN BOUND
================================================================================
15.1 WHAT WE NEED.
Fix n = 2m even, and let alpha = B / C_nor be the connection of 14.8.
Recall d alpha is a two-form on the simplex Delta_n, and A_h is its
matrix in the section t_0 = 0. Our goal:
E_{Delta_n} |Pf(A_h)| <= C_R Q_R^n prod_{i=1}^n l_i, (15.1)
for constants C_R, Q_R independent of n, g, and h. Once we have (15.1)
with Q_R < 1/M_p, the argument of 12.4 finishes the proof.
Actually, we need slightly more: we need the bound for each fixed tuple
h uniformly in the tuple, so we can sum over h in T_n(g).
15.2 DECOMPOSITION OF THE CURVATURE.
Differentiate
alpha = C_nor^{-1} (sum_{j<b_*} B_j + W B_{b_*}):
d alpha = C_nor^{-1} sum_{j<b_*} dB_j + C_nor^{-1} W dB_{b_*} + Q,
where
Q = -C_nor^{-2} dC_nor ^ B.
The last term Q is a sum of rank-2 wedges: Q ^ Q = 0, and its
coefficient bound is |Q_{ii'}| <= C_R l_i l_{i'}. In the Pfaffian
partition (section 15.5), Q can absorb at most 2 of the n indices; that
is, we can treat it as a constant-order perturbation.
The other b_* + 1 = S - 1 stages, with S = b_* + 2 total (B_j stages,
the B_{b_*} stage, and Q), contribute the principal part of the Pfaffian.
15.3 ESTIMATE FOR SMOOTH STAGES.
Fix 1 <= j <= b_* (including the terminal stage) and a tuple h in
T_n(g). The two-form dB_j on Delta_n is written as a pairing
(dB_j)_{ii'} = <X_i, T_j X_{i'}>
in an auxiliary Hilbert space H_j = L^2(d theta_j), where
d theta_j(u, y) = Y_j(u)_y du (mass form on (0,1) x Lambda_j),
X_i(u, y) = d_i Y_j(u)_y / Y_j(u)_y (or 0 where this diverges),
T_j(u, y; v, z) = sum_{a in Z} U_j(v + a - u) K_j(y, g^a z).
Key properties:
* T_j is skew-adjoint for the standard pairing (odd U_j, symmetric K_j,
simultaneous equivariance of K_j);
* the norms ||X_i||_{H_j} are bounded by a_j l_i / sqrt(delta_j), where
a_j = C epsilon_j^{-10} (1 <= j < J_*),
a_j = C r^{10} epsilon_j^{-10} (J_* <= j <= b_*).
This is exactly the Fisher estimate of the channel proposition,
applied along the direction (delta_{h_{i-1}} - delta_{h_i}) in the
smoothed probability at each time.
Eigenvalue estimate. Let E be the span of the X_i for i in a subset
I subset {1, ..., n} of size k = |I|. Let s_1 >= s_2 >= ... be the
singular values of the compression P_E T_j |_E. Then
|Pf_I(dB_j)| <= prod_{i in I} ||X_i||_{H_j}
* (prod_{nu=1}^k s_nu)^{1/2}. (15.3)
This is exactly the compression inequality of the original 6.2.
Consequence for 1 <= j < b_*. Here U_j has compact support, so T_j is
a Hilbert-Schmidt operator with
||T_j||_{HS}^2 <= C delta_{j+1}.
Consequently the product of the singular values is small:
(prod_nu s_nu)^{1/2} <= C^k delta_{j+1}^{k/4} k^{-k/4}.
Combining with ||X_i|| <= a_j l_i / sqrt(delta_j) and simplifying
delta_j^{-k/2} delta_{j+1}^{k/4} = beta_j^{-k/8} (n/k)^{k/4}, we get
|Pf_I(dB_j)| <= (C a_j beta_j^{-1/8})^k (n/k)^{k/4} prod_{i in I} l_i.
(15.4)
Consequence for j = b_*. The terminal T_{b_*} is not compactly
supported in time. Its key feature: it becomes (in the infinite-order
case) a "sign kernel" on each g-orbit of labels, and (in the finite-
order case) a sum of order kernels and their translations. Either way,
its compression onto any finite-dimensional E has singular values
s_nu <= C / nu for nu = 1, 2, 3, ... (15.5)
This is proved by approximating T_{b_*} with rank-N operators; the
error decays like C/N. For instance, in the infinite-order case, the
rank-N approximation comes from partitioning the line into N
subintervals of equal mass and replacing the sign kernel by a piecewise
constant on intervals. For the finite-order case, one uses the three
shifted pieces sign(v-u) * 1_{y=z}, 1_{v<u} * 1_{y=gz}, -1_{v>u} *
1_{y=g^{-1}z} and approximates each in the same way. Orthogonal
compression cannot increase rank or error.
Inserting (15.5), together with W^{k/2} from the multiplier and using
k! >= (k/e)^k, we get
|Pf_I(W dB_{b_*})| <= (C a_{b_*} sqrt(W) beta_{b_*}^{-1/2})^k
* (n/k)^{k/2} prod_{i in I} l_i. (15.6)
15.4 ESTIMATE FOR THE RAW STAGE.
The raw stage B_0 has curvature
(dB_0)_{ii'} = sum_{a in Z} U_0(t_{i'} + a - t_i) K_0(J_i, g^a J_{i'}),
where J_i = delta_{h_{i-1}} - delta_{h_i},
K_0(h, h') = chi_R(d(h, h')), and l_i, l_{i'} are as usual. Since chi_R
is Lipschitz with constant 1/R, we get
|K_0(J_i, g^a J_{i'})| <= 2 l_i l_{i'} / R.
Also, U_0 vanishes outside |s| <= 2 R / n, so the (i, i') entry is
nonzero only when t_i, t_{i'} are within 2R/n modulo one.
Hadamard bound. For an even |I| = k, put
N_i(I) = #{i' in I : |t_i - t_{i'}|_{circle} <= 2R/n OR i' = i}.
Then
|Pf_I(dB_0)| <= (C/R)^{k/2} prod_{i in I} l_i
* prod_{i in I} N_i(I)^{1/4}. (15.7)
This is the "each row has at most sqrt(N_i) nonzero entries of size
1/R" bound, converted to a Pfaffian via |Pf(A)|^2 = |det(A)| and
Hadamard.
Sparsity lemma (occupancy). Choose the ordered coordinate tuple
(t_1, ..., t_n) uniformly on Delta_n and independently choose a
k-element subset I of {1, ..., n}. Then
E_{Delta_n, I} prod_{i in I} N_i(I)^{1/4} <= C^k R^{k/4}. (15.8)
Idea: uniform simplex coordinates are order statistics of n independent
uniform points on the circle. Selecting a uniform k-subset is
equivalent to picking k of those points uniformly at random. Partition
the circle into m' = floor(n/(2R)) equal cells; a point can only see
its neighbors in adjacent cells. Using multinomial combinatorics on the
cell occupancies gives the stated bound, up to a small factor.
Combining (15.7) and (15.8) gives a factor of (C R^{-1/4})^k for the
raw stage, after averaging the tuple weights over the simplex.
15.5 EXPONENT BOOKKEEPING AND THE FINAL PFAFFIAN BOUND.
Now we tie everything together. Each of the S = b_* + 2 stages
contributes a "base" of the form
base_j =
(C a_j beta_j^{-1/8})^k (1 <= j < J_*)
(C a_j beta_j^{-1/8})^k (J_* <= j < b_*)
(C a_{b_*} sqrt(W) beta_{b_*}^{-1/2})^k (j = b_*) (15.9)
(C R^{-1/4})^k (raw stage j = 0)
C_R (times (n/k)^{-k/2} entropy factor) (Q, at most rank-2).
Plug in the parameter choices from 14.1:
* for 1 <= j < J_*: a_j <= C (bounded independent of R), beta_j >= R,
so base_j <= C R^{-1/8};
* for J_* <= j < b_*: a_j <= C R^{10} (1 + b_*)^{20} by the later
Fisher bound and epsilon_j >= epsilon_* / (1 + b_*)^2, while
beta_j >= R^{200}. So base_j <= C R^{-15} (1 + b_*)^{20};
* at j = b_*: beta_{b_*} >= W^2 R^{100}, so sqrt(W) / beta_{b_*}^{1/2}
<= 1 / R^{50}, hence base_{b_*} <= C R^{-40} (1 + b_*)^{20};
* raw stage: C R^{-1/4}.
For R sufficiently large, all of these bases are at most
Q_R = C_1 R^{-1/10} (15.10)
for a fixed constant C_1 (enlarge C_1 as needed; it depends only on the
word metric and the fixed construction choices).
Pfaffian partition expansion. We want a bound on |Pf of the full
d alpha|. Write d alpha as a sum of S two-forms M_0, ..., M_{S-1}
(raw stage, b_*+1 smooth stages, Q). Then
Pf(sum M_l) = sum over ordered partitions I_0 (+) ... (+) I_{S-1}
of {1, ..., n}, all sizes even,
sigma(I_0, ..., I_{S-1}) prod_l Pf_{I_l}(M_l), (15.11)
with sigma = +-1. Each partition term, at a given point of Delta_n,
is bounded by the product of the individual Pf_I estimates. Taking
absolute values and averaging over Delta_n:
E |Pf(d alpha)| <= sum over partitions of sized ((k_l))
* (number of such partitions) * prod_l (bound at
size k_l).
Routine bookkeeping:
* The number of ordered partitions of sizes (k_l) is n!/(prod_l k_l!),
summing over k_l to at most S^n.
* The "shape" factor (n/k_j)^{k_j/2} from every smooth stage,
multiplied together, contributes at most S^{n/2} (by convexity of
p |-> p log(1/p) on the simplex of proportions).
* The Q stage absorbs at most 2 indices, so its contribution is
C_R times Q_R^{-2}, absorbed into C_R.
* The raw stage's average (15.8) is absorbed by the R^{-1/4} base.
Putting these together:
E_{Delta_n} |Pf(d alpha)| <= C_R (Q_R S^{3/2})^n prod_i l_i,
(15.12)
where S = b_* + 2. This is the "curvature bound" of the paper.
15.6 THE KEY INPUT: b_* = O(1 + log R).
From the recursion beta_{j+1} = beta_j^{3/2}, log beta_j grows like
(3/2)^{j-1} log R. Solving for beta_{b_*} >= W^2 R^{100}:
(3/2)^{b_*-1} log R >= 2 C_* r + 100 log R,
so b_* = O(1 + log R) as R -> infinity (r is a fixed multiple of R).
Hence S = O(1 + log R), and the base in (15.12) is
Q_R S^{3/2} = C_1 R^{-1/10} * O((log R)^{3/2}) for large R.
Since R^{-1/10} beats any power of log R, we have
Q_R S^{3/2} -> 0 as R -> infinity.
Taking R large enough closes the whole curvature estimate.
================================================================================
16. STEP 5 CONTINUED: PROOF OF THE MAIN THEOREM
================================================================================
16.1 THE SINGLE NUMERICAL CHOICE.
Fix, once and for all, a spatial radius R large enough that:
* R satisfies all the hypotheses of sections 14-15 (so the connection
construction and the curvature estimate apply);
* the base Q_R of (15.10) satisfies Q_R <= (1/2) / M_p.
Here M_p = sum_w |p(w)|_1 (1 + |w|) is the exponential moment from
section 10.7, which is fixed by the input p alone. Since Q_R S^{3/2}
tends to zero as R -> infinity (15.6), such an R exists.
This fixes:
* the spatial radius r = 6^{J_*-1} R;
* the ladder beta_j;
* the stage count b_* = O(1 + log R);
* the multiplier W and the scales delta_j / n;
* every constant C_R below.
16.2 THE TRACE VANISHES AT INFINITE-ORDER g.
Let g be an element satisfying the twist assumption of (14.4):
(i) either g has infinite order, or
(ii) g has finite order and min_h d(h, gh) > 4r.
Under this assumption, the connection construction and the curvature
estimate apply. Fix any even n > 10 beta_{b_*} and consider the trace
formula (12.3):
sum_h v_{n+1}(h) E_{Delta_n} Pf(A_h) = (-1)^m trace_[g](p).
Take absolute values, insert the curvature bound (15.12), and then the
moment bound (11.3):
|trace_[g](p)|
<= sum_h |v_{n+1}(h)| * C_R (Q_R S^{3/2})^n prod_i l_i
<= C_R (Q_R S^{3/2})^n * M_p^{n+1}
<= C_R M_p (Q_R M_p)^n * S^{3n/2}.
Only the last line changed: (Q_R S^{3/2}) is replaced by (Q_R M_p S^{3/2}),
and since S^{3/2} is O(polylog(R)) and Q_R M_p < 1/2, the factor
Q_R M_p S^{3/2} is also < 1 for our fixed R (enlarge R if necessary).
Write it as rho < 1.
Then |trace_[g](p)| <= C_R M_p rho^n for every even n. Taking n ->
infinity through even integers gives trace_[g](p) = 0.
So the trace vanishes at every g satisfying the twist assumption.
16.3 THE SURVIVING FINITE-ORDER CLASSES.
By 16.2, an element g of finite order can fail the twist assumption only
if
min_h d(h, gh) <= 4r,
that is, some conjugate of g has word length at most 4r. Since the word
metric is left-invariant,
d(h, gh) = |h^{-1} g h|,
so failure of the twist assumption for g is equivalent to the existence
of h with h^{-1} g h in the ball B(1, 4r).
Therefore any finite-order conjugacy class [g] with trace_[g](p) != 0
intersects the fixed finite ball B(1, 4r). The ball is finite (the group
is finitely generated), so only finitely many finite-order classes can
have nonzero trace.
Combining with 16.2, we conclude:
For p with exponential decay, the trace trace_C(p) is supported on a
finite subset F of the set C_fin(G) of finite-order classes.
This is Theorem 8.1 for the finitely generated, exponentially decaying
case.
16.4 TRANSFER TO THE ORIGINAL e AND G.
By Proposition 10.6, we may assume without loss of generality that:
* G is finitely generated;
* e has exponential decay.
For a general discrete group G' containing G as a finitely generated
subgroup (which is the setting of Proposition 10.6), the trace of the
original idempotent e in M_q(l^1(G')) is a finite sum of
trace coefficients of p over conjugacy classes of G that are contained
in classes of G'. Since:
* every finite-order class of G' that meets G is a union of
finite-order classes of G,
* every infinite-order class of G' that meets G is a union of
infinite-order classes of G, each with trace = 0,
the finite support result transfers from G to G'.
Finally, an arbitrary element of K_0(l^1(G')) is a difference [e] - [e']
of two idempotent classes (section 6). The image
HS^1([e] - [e']) is supported on the union of the two finite sets of
classes obtained from 16.3, which is again finite.
This proves the theorem:
Theorem (l^1-Bass, every discrete group). Let G be any discrete group,
let q >= 1, and let e in M_q(l^1(G)) be an idempotent. Then there is
a finite set F of finite-order conjugacy classes of G such that
trace_C(e) = 0 for every C not in F.
================================================================================
APPENDIX A. GLOSSARY
================================================================================
A.1 GROUPS AND THE WORD METRIC.
group a set with multiplication, identity, inverse,
associativity.
discrete group a group with no topological structure (any group,
considered abstractly).
identity e the neutral element of the group.
inverse g^{-1} the element satisfying g g^{-1} = g^{-1} g = e.
conjugacy class the equivalence class [g] = {x g x^{-1} : x in G}
under the equivalence g ~ h iff h = x g x^{-1}.
centralizer C_G(g) = {x in G : x g x^{-1} = g}. Equals the
stabilizer of g under conjugation.
order of g the least m >= 1 with g^m = e, or infinity.
finite-order having finite order. A finite-order class is a class
whose elements have finite order.
word metric d(x, y) = |x^{-1} y|_S for a fixed finite symmetric
generating set S, where |.|_S is the shortest word
length spelling an element.
ball B(x, s) = {y : d(x, y) <= s}. Growth controlled by
(1.4.1): |B(x, s)| <= exp(c_d s).
A.2 GROUP ALGEBRAS.
complex group ring C[G]: finite formal combinations
sum_{g in G} a_g g with a_g in C.
l^1 group algebra l^1(G): all functions a: G -> C with
||a||_1 = sum_g |a(g)| < infinity.
convolution (a * b)(g) = sum_{h k = g} a(h) b(k).
submultiplicativity ||a * b||_1 <= ||a||_1 ||b||_1.
matrix algebra M_q(l^1(G)): q x q matrices over l^1(G), with
||A||_1 = sum_{i,j,g} |A_{ij}(g)|.
Banach algebra complete normed algebra with submultiplicative
norm. Both l^1(G) and M_q(l^1(G)) are Banach.
Neumann series (1 - b)^{-1} = sum_{k>=0} b^k whenever
||b|| < 1.
A.3 IDEMPOTENTS AND MODULES.
idempotent p with p^2 = p.
projective module direct summand of a free module. Equivalently,
p A^q for some idempotent p and some q.
Serre-Swan picture in our setting: "projective module" = "image of
an idempotent matrix".
similar idempotents p and p' such that p' = u p u^{-1} for some
invertible u. They define isomorphic modules.
A.4 TRACES AND K-THEORY.
ordinary matrix trace sum_{i=1}^q p_{ii} (does not descend to modules).
Hattori-Stallings trace_C(p) = sum_{g in C} sum_{i=1}^q p_{ii}(g).
Well-defined on K_0. Independent of representative
of the module.
K_0(A) the Grothendieck group of finitely generated
projective right A-modules; generated by [P] with
[P (+) P'] = [P] + [P'].
HS^1 the map HS^1: K_0(l^1(G)) -> l^1(C(G)), sending
[p] |-> (trace_C(p))_C.
A.5 STATEMENT OF CONJECTURES.
Bass conjecture (1976, algebraic, Z[G]) trace vanishes on all
classes other than [e].
complex Bass (over C[G]) trace vanishes on infinite-order
classes and all but finitely many finite-order
classes.
l^1-Bass (BCM2004) (over l^1(G)) trace vanishes on infinite-order
classes and all but finitely many finite-order
classes.
Main theorem the l^1-Bass statement, proved here for every
discrete group.
A.6 ANALYTIC CORRECTION.
exponential decay sum_{h in Gamma} |p(h)|_1 exp(c |h|) < infinity
for some c > 0 in a finitely generated subgroup
Gamma.
moment M_p sum_w |p(w)|_1 (1 + |w|); finite under exponential
decay.
analytic correction replacing an l^1 idempotent e by a similar
exponentially decaying idempotent p = (1/2) I +
(a - I/2) Q with a close to e, b = 4(a - a^2),
Q = (1 - b)^{-1/2}.
A.7 CYCLIC TUPLES AND WEIGHTS.
cyclic tuples T_k(g) H-orbits in G^{k+1}, where H = C_G(g), acting by
simultaneous left multiplication. Tuple h =
(h_0, ..., h_k) indexed modulo k+1 with closing
step h_{i+k+1} = g h_i.
tuple weights v_{k+1}(h) = tr(prod_{i=0}^k p(h_i^{-1} h_{i+1}))
where the product and the matrix trace are
ordinary.
edge lengths l_i = 1 + d(h_{i-1}, h_i), with periodic
indexing.
weight identity trace_[g](p) = sum_h v_{k+1}(h); rotation and
deletion (contraction of one factor against p)
are compatible.
moment bound sum_h |v_{k+1}(h)| prod_i l_i <= M_p^{k+1}.
A.8 SIMPLICES AND DIFFERENTIAL FORMS.
simplex Delta~_k ordered times 0 <= t_0 <= ... <= t_k <= t_0 + 1.
gap coordinates x_i = t_{i+1} - t_i, sum x_i = 1, x_i >= 0.
quotient simplex Delta_k = {0 <= t_1 <= ... <= t_k <= 1} (section
t_0 = 0); same object, differing in normalization.
translation vector V sum_{i=0}^k d/dt_i (times move all t_i together).
face inclusion x_i = 0 identifies with the simplex of the
shortened tuple.
Pfaffian for a skew-symmetric 2m x 2m matrix A:
Pf(A)^2 = det(A), Pf(A) dt_1 ^ ... ^ dt_{2m}
= (1/m!) (sum A_{ij} dt_i ^ dt_j)^m.
A.9 CONNECTIONS AND CURVATURE.
connection family assignment h |-> alpha_h (a one-form on
Delta~_k) invariant under time translation,
centralizer-equivariant, rotation-equivariant,
face-compatible, smooth with polynomial bounds.
curvature d alpha_h; a basic (translation-invariant)
two-form on Delta~_k.
trace-as-curvature sum_h v_{n+1}(h) E Pf(A_h) = (-1)^m trace_[g](p).
Chern-Simons form eta = (alpha - alpha') ^ sum_{j} (d alpha)^j ^
(d alpha')^{m-1-j}; transgression between two
choices of connection.
A.10 CHANNELS.
concentration chi_r chi_r(s) = min(1, max(0, 2 - s/r)).
score E_r(P, y) sum_h P_h chi_r(d(h, y)).
max score q_r(P) max_y E_r(P, y).
profile phi_r(P) (q_r(P) - a_*)_+ with a_* = 0.80.
channel left-equivariant smooth map from finitely supported
probabilities on G to subprobabilities on an
auxiliary label set.
selector A smooth softmax that picks out high-scoring centers.
distribution D like A but with lower thresholds, ensures support
near a maximizing center.
prediction L channel built from D, used with a fixed kernel
kappa to estimate scores at readers.
Fisher seminorm F_Y(P; h, h') = (sum_omega |dot Y(P)_omega|^2 /
Y(P)_omega)^{1/2}, controlling derivatives of the
channel with respect to the input probability.
A.11 THE AVERAGING CONSTRUCTION.
stages and scales delta_1 < ... < delta_{b_*}, chosen as beta_j/n
with beta_1 = R, beta_{j+1} = beta_j^{3/2}.
smoothed probability P_j(u) = (mu_j * ... * mu_1 * delta_{f(u)})(u),
a time-averaged version of the tuple step
function.
lag kernel U_j odd function supported in
[-2 delta_{j+1}, 2 delta_{j+1}], satisfying
U_j' = 2(delta_0 - rho_j) for rho_j = mu_{j+1} *
mu_{j+1}.
stage form B_j -(1/2) int int U_j(v - u) K_j(dY_j(u), Y_j(v)).
aggregate B sum_j B_j + W B_{b_*}, with W = exp(C_* r).
contraction B(V) B contracted with the translation vector;
compute C_nor.
positivity C_nor >= c_0 = 1/10 because of telescoping
C_0 >= 1/5 - Phi_1 and
C_j >= Phi_j - Phi_{j+1} - 30 epsilon_j and
W C_{b_*} >= Phi_{b_*}.
normalized connection alpha = B / C_nor (a fixed positive
alpha(V) = 1.
A.12 CURVATURE ESTIMATE.
weighted Hilbert H_j = L^2(d theta_j) for
space d theta_j(u, y) = Y_j(u)_y du.
kernel T_j T_j(u, y; v, z) = sum_a U_j(v + a - u) K_j(y, g^a
z); skew-adjoint.
channel derivative ||X_i||_{H_j} <= a_j l_i / sqrt(delta_j), where
bound a_j = C epsilon_j^{-10} (early) or
C r^{10} epsilon_j^{-10} (late).
compression bound |Pf_I(dB_j)| <= prod_{i in I} ||X_i|| *
(prod_nu s_nu)^{1/2}.
raw stage |Pf_I(dB_0)| <= (C/R)^{k/2} prod_i N_i(I)^{1/4}
prod_i l_i.
occupancy average E_{Delta_n, I} prod_i N_i(I)^{1/4} <= C^k R^{k/4}.
final curvature bound E |Pf(d alpha)| <= C_R (Q_R S^{3/2})^n prod_i l_i,
with Q_R = C_1 R^{-1/10} and S = b_* + 2.
A.13 CLOSING THE ARGUMENT.
|trace_[g](p)| <= C_R M_p (Q_R M_p S^{3/2})^n -> 0 as n ->
infinity through evens, for R fixed large enough.
finite-order g twist assumption (14.4) fails only if some
conjugate has word length <= 4r; finitely many
such classes.
K_0 arbitrary element is a difference of two
idempotent classes, whose supports are unions of
two finite sets.
================================================================================
APPENDIX B. FURTHER READING
================================================================================
B.1 SUGGESTED ORDER.
1. Hattori (1965) or Stallings (1965) for the original trace. Short
algebraic papers; the trace formula on page one is what we have
generalized.
2. Bass (1976) for the algebraic conjecture. Section 1 explains why
one might hope for the conjecture to be true.
3. Burghelea (1985) for the cyclic-homology route. Our sections 11-12
mirror the structure of that computation, in a form adapted to l^1.
4. Berrick-Chatterji-Mislin (2004) for the l^1 statement, the
Bost-assembly approach, and the specific conjecture this document
proves.
5. Ji-Ogle-Ramsey (2010, 2014) for rapid-decay completions, which
motivated some of the exponential-decay structure in our argument.
6. Emmanouil (1998) for the cyclic-periodicity approach to the
complex-group-ring version, which is another route to the trace
support theorem.
B.2 WHERE THE ARGUMENT SITS.
The idea of using cyclic tuples and integral forms is old (Burghelea,
Eckmann), but the l^1 completion opens up a smoothing argument that is
not available over C[G]. The multiscale averaging construction in
section 14 is essentially: "instead of trying to solve the problem
exactly at one radius, resolve it across a stack of increasing time
scales, where each intermediate stage is approximable by a small
finite-dimensional sample." That idea shows up in other guises in
topological dynamics and in the theory of trace class operators; the
application to Bass's conjecture is the novelty.
B.3 HOW TO FIND THE PAPER THIS DOCUMENT SUMMARIZES.
The original multilanguage LaTeX sources are in the repository
"ezmath2" under the directory
The-l1-Bass-Conjecture-for-Discrete-Groups/. The detailed proof of
every lemma quoted here is there. The present document is meant to be
read alongside the original, not instead of it.
================================================================================
APPENDIX C. REFERENCES
================================================================================
Hattori, A. Rank element of a projective module. Nagoya Math. J. 25
(1965), 113-120.
Stallings, J. Centerless groups--an algebraic formulation of Gottlieb's
theorem. Topology 4 (1965), 129-134.
Bass, H. Euler characteristics and characters of discrete groups.
Inventiones Math. 35 (1976), 155-196.
Burghelea, D. The cyclic homology of the group rings. Comment. Math.
Helv. 60 (1985), 354-365.
Eckmann, B. Cyclic homology of groups and the Bass conjecture.
Comment. Math. Helv. 61 (1986), 193-202.
Emmanouil, I. On a class of groups satisfying Bass' conjecture.
Inventiones Math. 132 (1998), 307-330.
Lafforgue, V. K-theorie bivariante pour les algebres de Banach et
conjecture de Baum-Connes. Inventiones Math. 149 (2002), 1-95.
Berrick, A.J., Chatterji, I., Mislin, G. From acyclic groups to the
Bass conjecture for amenable groups. Math. Ann. 329 (2004), 597-621.
Ji, R., Ogle, C., Ramsey, B. Relatively hyperbolic groups, rapid decay
algebras and a generalization of the Bass conjecture. J. Noncomm.
Geom. 4 (2010), 83-124.
Ji, R., Ogle, C., Ramsey, B.W. On the Hochschild and cyclic
(co)homology of rapid decay group algebras. J. Noncomm. Geom. 8
(2014), 45-59.
Hoeffding, W. Probability inequalities for sums of bounded random
variables. J. Amer. Statist. Assoc. 58 (1963), 13-30.
Chern, S.-S., Simons, J. Characteristic forms and geometric invariants.
Ann. of Math. 99 (1974), 48-69.
Engel, A., Marcinkowski, M. Burghelea conjecture and asymptotic
dimension of groups. J. Topol. Anal. 12 (2020), 321-356.
OpenAI. The Bass trace conjecture and the characteristic-zero Kaplansky
idempotent conjecture. OpenAI Math Release preprint, September 2026.
(Available in the same GitHub repository as the present document.)
================================================================================
END OF DOCUMENT
================================================================================