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THE l^1-BASS CONJECTURE FOR DISCRETE GROUPS
A complete, self-contained account for a curious high-school reader
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Author: OpenAI Date: October 2026
Rewritten to be clearer, more correct, more concise, more approachable.
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HOW TO READ THIS DOCUMENT
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This is a complete proof that the l^1-Bass conjecture holds for every
discrete group. Every step is written out. Every symbol is defined before
it is used. Each substantial lemma has a short "idea" paragraph that says
WHY it is true, not just THAT it is true.
Prerequisites assumed:
* you know what a group is (section 1 reminds you);
* you are comfortable with complex numbers and matrices;
* you are willing to follow a few sums and integrals.
Nothing about K-theory, algebraic topology, differential geometry, or
functional analysis is required: these objects appear only when needed,
and each is introduced with enough examples that a motivated reader
could invent them.
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 ||ยท||_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".