arXiv:1609.07635v1 [] 24 Sep 2016

AMENABLE GROUPS OF FINITE COHOMOLOGICAL DIMENSION
AND THE ZERO DIVISOR CONJECTURE
DIETER DEGRIJSE
Abstract. We prove that every amenable group of cohomological dimension two whose
integral group ring is a domain is solvable and investigate certain homological finiteness
properties of groups that satisfy the analytic zero divisor conjecture and act on an acyclic
CW -complex with amenable stabilisers.

1. Introduction
All groups appearing in this paper are assumed to be discrete. Recall that the class
of elementary amenable groups is by definition the smallest class of groups that contains
all finite and all abelian groups and that is closed under directed unions, extensions,
quotients and subgroups. A group G is said to be amenable if for every continuous action
of G on a compact, Hausdorff space X, there is a G-invariant probability measure on X.
As the name suggest, all elementary amenable groups are amenable (e.g. see [3, Appendix
G]) and by now there is a multitude of examples of groups that are amenable but not
elementary amenable. Such groups arise for example in the class of branch groups, which
includes the first example by Grigorchuk of a group of intermediate growth (see [2],[17]).
Another important source of amenable non-elementary amenable groups are (commutator
subgroups of) topological full groups of Cantor minimal systems (see [16]) and we refer
to the introduction of [17] for more examples.
However, as far as we are aware none of the currently known examples of amenable
but not elementary amenable groups are known to have finite cohomological dimension
over any field. For example, many Branch groups, including Grigorchuk’s example, are
commensurable to an n-fold direct product of themselves for some n ≥ 2 (see [2]). By
Corollary 2.3, the cohomological dimension of such a non locally-finite group is infinite
over any field. Since
L commutator subgroups of topological full groups of Cantor minimal
systems contain n≥0 Z as a subgroup (see [13, Remark 4.3(2)]), these groups also have
infinite cohomological dimension over any field. Hence, one might well wonder if in fact all
amenable groups of finite cohomological dimension (over some ring or field) are elementary
amenable.
We will be working with cohomological dimension over the integers, which we will
therefore just refer to as cohomological dimension. Since all elementary amenable groups
of finite cohomological dimension must be torsion-free and have finite Hirsch length by
1

2

[14, Lemma 2], one can conclude from [15, Corollary 1] that they are virtually solvable.
We are therefore led to the following question, which was brought to the authors attention
by Peter Kropholler.
Question. Is every amenable group of finite cohomological dimension virtually solvable?
By Stalling’s theorem ([27]) this question obviously has a positive answer for groups of
cohomological dimension at most one. By the Tits alternative, all amenable linear groups
of finite cohomological dimension are virtually solvable. Also note that the answer to the
more general question where one replaces ‘amenable’ by ‘does not contain non-abelian
free subgroups’ is no. Indeed, Olshanskii’s example of a torsion-free Tarski monster, i.e. a
finitely generated torsion-free non-cyclic group all of whose proper subgroups are cyclic, is
non-amenable and admits an aspherical presentation, showing that it has cohomological
dimension two (e.g. see [25]).
Our main results, which we will state below, depend on the validity of Kaplansky’s
zero divisor conjecture for group rings or a generalisation of it due to Linnell (e.g. see
[24]), called the analytic zero divisor conjecture, which states that if α ∈ C[G] \ {0} and
β ∈ l2 (G) \ {0}, then αβ 6= 0. There are no known counterexamples to the (analytic) zero
divisor conjecture and it has been proven for a wide class of groups, including torsion-free
elementary amenable groups (see [19, Th. 1.4]) and extensions of right orderable groups by
torsion-free elementary amenable groups (e.g. see [24, Th. 8.2]). Moreover, Elek showed
in [11] that for finitely generated amenable groups the Kaplansky zero divisor conjecture
over C and the analytic zero divisor conjecture are equivalent. We also remark that for a
torsion-free group G, Z[G] is a domain if and only if Q[G] is a domain and for any field
Z ⊆ k ⊆ C, the Atiyah conjecture of order Z implies that k[G] is a domain (e.g. see [23,
Lemma 10.15]).
Our first main result concerns the homological finiteness properties F Pn and F P of a
group. These notions will be recalled in the next section.
Theorem A. Let G be an amenable group such that Z[G] is a domain. If G has cohomological dimension n and is of type F Pn−1 , then G is of type F P .
Note that the conclusion of Theorem A is certainly false in general for non-amenable
torsion-free groups. Indeed, for n ≥ 1, the Bestvina-Brady group HL associated to any
flag-triangulation L of the (n − 1)-sphere contains a non-abelian free subgroup, has cohomological dimension n by [21, Th. 22] and is of type F Pn−1 but not of type F P by the
main theorem of [5]. Also, the Tarski monster mentioned above is finitely generated and
of cohomological dimension two, but is not finitely presented.
Using Theorem A we obtain a positive answer to the question above in the 2-dimensional
case, assuming the validity of the zero-divisor conjecture. This generalises [18, Th. 3].

3

Theorem B. Every amenable group G of cohomological dimension 2 such that Z[G] is a
domain is solvable and hence isomorphic to a solvable Baumslag-Solitar group BS(1, m)
for some non-zero m ∈ Z or to a non-cyclic subgroup of the additive rationals.
A modified version of the proof of Theorem A gives the following theorem.
Theorem C. Let G be a group of cohomological dimension n that satisfies the analytic
zero-divisor conjecture and admits an acyclic G-CW-complex X with amenable stabilisers.
If G is of type F Pn−1 and X does not have a free G-orbit of n-cells, then G is of type F P .
Since an n-dimensional G-CW-complex obviously has no free G-orbit of (n + 1)-cells,
the following corollary is immediate.
Corollary 1. Let Γ be a group that admits an acyclic n-dimensional Γ-CW-complex with
amenable stabilisers. Then any subgroup G of Γ of cohomological dimension n + 1 that
satisfies the analytic zero-divisor conjecture and is of type F Pn is of type F P .
Recall that a group is said to be almost coherent, if every finitely generated subgroup
is almost finitely presented, i.e. of type F P2 .
Corollary 2. Fundamental groups of graphs of groups with cyclic edge groups and vertex
groups that are either subgroups of the additive rational numbers or solvable BaumslagSolitar groups are almost coherent if they satisfy the analytic zero divisor conjecture.
A group G is said to commensurate a subgroup H if H g ∩ H has finite index in both H
and H g for every element g ∈ G. In particular, if H is normal in G then G commensurates
H. Using Theorem C, we prove the following.
Corollary 3. Let G be a group that satisfies the analytic zero-divisor conjecture and
commensurates a non-trivial amenable subgroup. If G has cohomological dimension n and
is of type F Pn−1 , then G is of type F P .
Acknowledgements. The author is grateful to Peter Kropholler for introducing him
to the question posed in this paper and for valuable discussions and remarks concerning
a preliminary version of this work. This research was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation
(DNRF92).
2. Finiteness conditions
We start by recalling some basic facts about homological finiteness properties of groups
and refer the reader to [6], [8] and [31] for more details. Let G be a group, let R be a
commutative ring with unit and let R[G] denote the group ring of G with coefficients in
R. In this paper we will always consider left R[G]-modules, unless stated otherwise. If
there is no mention of a ring R, then R is assumed to equal Z.
The cohomological dimension cdR (G) of G over R is by definition the length of the
shortest projective R[G]-resolution of the trivial R[G]-module R. If there does not exist

4

such a finite length projective resolution, then the cohomological dimension is by definition
infinite. Equivalently, cdR (G) is the largest integer n ≥ 0 for which there exists an R[G]module M such that the n-th cohomology group Hn (G, M) is non-zero. The homological
dimension hdR (G) of G over R is by definition the length of the shortest flat R[G]resolution over the trivial R[G]-module R. Again, if there does not exist a finite length flat
resolution then the homological dimension is by definition infinite. Equivalently, hdR (G)
is the largest integer n ≥ 0 for which there exists a right R[G]-module M such that the
n-th homology group Hn (G, M) is non-zero. In general one has hdR (G) ≤ cdR (G), and
if G is countable one also has cdR (G) ≤ hdR (G) + 1 (e.g. see [6, Th 4.6]). Note that
cdR (G) ≤ cd(G), and that cd(G) < ∞ implies that G is torsion-free.
The group G is said to be of type F Pn over R for an integer n ≥ 0 if and only if
there exists a projective R[G]-resolution P∗ of the trivial R[G]-module R such that Pd is
a finitely generated R[G]-module for all 0 ≤ d ≤ n. The group G is said to be of type F P
over R if and only if there exists a finite length projective R[G]-resolution P∗ of the trivial
R[G]-module R such that Pd is a finitely generated R[G]- module for all d ≥ 0. Every
group G is of type F P0 over R and G is of type F P1 over R if and only if G is finitely
generated. If G is finitely presented then G is of type F P2 over R but not necessarily the
other way around (see [6, Prop. 2.1 and 2.2] and [5]). Note also that if G is of type F Pn
over R and cdR (G) = n, then G is of type F P over R (e.g. see [8, VIII Prop. 6.1]).
The following lemma is a variation on a classical criterion due to Strebel implying that
a group is of type F Pn . It will be well-known to experts.
Lemma 2.1. Let R be a commutative ring with unit and let G be a group with cdR (G) = n
that is type F Pn−1 over R. Then G is of type F P over R if and only if for any directed
system of free R[G]-modules {Fi }i∈I with lim Fi = 0 one has
−→
n
lim H (G, Fi ) = 0.
−→
Proof. A classical result of Strebel (e.g. see [6, Th. 1.3.]) implies that if G is of type
F Pn−1 over R and cdR (G) = n then G is of type F P if and only if lim Hn (G, Mi ) = 0
−→
for any directed system of R[G]-modules {Mi }i∈I with lim Mi = 0. Let {Mi }i∈I be such
−→
a directed system. Now for each i ∈ I, consider the free R[G]-module
M
Fi =
R[G]
m∈Mi \{0}

and the surjection of R[G]-modules
πi : Fi → Mi : em 7→ m
where em denotes the identity element in the R[G]-factor corresponding to m ∈ M. Given
i ≤ j in I we have the corresponding R[G]-homomorphism ϕi,j : Mi → Mj . Define the

5

R[G]- homomorphism
θi,j : Fi → Fj : em 7→

n

eϕi,j (m) if ϕi,j (m) 6= 0
0
otherwise.

One checks that the maps θi,j assemble to form a directed system {Fi }i∈I with vanishing
direct limit and that
{πi }i∈I : {Fi }i∈I → {Mi }i∈I
is a surjective map of directed systems, i.e. ϕi,j ◦ πi = πj ◦ θi,j for all i ≤ j in I. Since
cdR (G) = n we obtain from a collection of natural long exact cohomology sequences, a
surjective map of directed systems
{Hn (G, Fi )}i∈I → {Hn (G, Mi )}i∈I → 0.
Since taking direct limits is exact, there is a surjection
lim Hn (G, Fi ) → lim Hn (G, Mi ) → 0,
−→
−→
from which the lemma easily follows.

Recall that two groups are said to be abstractly commensurable if they have isomorphic
subgroups of finite index. In the introduction we mentioned that non locally-finite groups
G that are abstractly commensurable to a direct product Gn for some n ≥ 2, have infinite
cohomological dimension over any field k. The following lemma will allow us to prove this
in the corollary below.
Lemma 2.2. Let k be a field and let Gi be a group with hdk (Gi ) = ni for i = 1, 2, then
hdk (G1 × G2 ) = n1 + n2 .
Proof. Let Mi be a right k[Gi ]-module such that Hni (Gi , Mi ) 6= 0 for i = 1, 2 and consider
right the k[G1 ×G2 ]-module M1 ⊗k M2 . By the Lyndon-Hochschild-Serre spectral sequence
associated to the extension
1 → G1 → G1 × G2 → G2 → 1
it suffices to show that
Hn2 (G2 , Hn1 (G1 , M1 ⊗k M2 )) 6= 0.
Let P∗ be a projective k[G1 ]-resolution of the trivial k[G1 ]-module k. Since
(M1 ⊗k M2 ) ⊗k[G ] P∗ ∼
= (M1 ⊗k[G ] P∗ ) ⊗k M2
1

1

and − ⊗k M2 is an exact functor because k is a field, we have
Hn (G1 , M1 ⊗k M2 ) ∼
= Hn (G1 , M1 ) ⊗k M2 .
1

1

A similar reasoning shows that
Hn2 (G2 , Hn1 (G1 , M1 ) ⊗k M2 ) ∼
= Hn1 (G1 , M1 ) ⊗k Hn2 (G2 , M2 ).

6

We conclude that
Hn2 (G2 , Hn1 (G1 , M1 ⊗k M2 )) ∼
= Hn1 (G1 , M1 ) ⊗k Hn2 (G2 , M2 ) 6= 0,
proving the lemma.

Corollary 2.3. Let G be a group that is not locally finite. If G is abstractly commensurable
to a direct product Gn for some n ≥ 2, then hdk (G), and hence also cdk (G), is infinite
for any field k.
Proof. Suppose hdk (G) < ∞. Then hdk (Gn ) < ∞ and we have hdk (G) = hdk (K) and
hdk (Gn ) = hdk (L) for every finite index subgroup K of G and finite index subgroup L
of Gn by [6, Cor. 5.10]. Since G is abstractly commensurable with Gn , this implies that
hdk (G) = hdk (Gn ). But then the lemma above says that n(hdk (G)) = hdk (Gn ) = hdk (G)
for some n ≥ 2, which implies that hdk (G) = 0. We now conclude from [6, Prop. 4.12(b)]
that G is locally finite, which is a contradiction. This proves that hdk (G) = ∞.

3. Non-commutative localisation
Later on we will need to embed the group ring Z[G] of a certain group G, into a ring
D such that every non-zero element of Z[G] is invertible in D. In this section we recall
how to construct such a ring D, assuming G satisfies the assumptions of either Theorem
A or Theorem B.
We start by briefly reviewing the Ore condition and the process of Ore localisation and
refer the reader to [20, 4.10] for details and proofs. Let R be a ring with a unit and let S
be a multiplicatively closed subset of R that contains the unit of R but does not contain
any zero divisors of R. The set S is said to satisfy the left Ore condition with respect to
R if for every s ∈ S and every a ∈ R, there exists t ∈ S and b ∈ R such that ta = bs. If
S satisfies the left Ore condition with respect to R one can construct the ring
S −1 R = S × R/ ∼
where (s, a) ∼ (s′ , a′ ) if there exist b, b′ ∈ R such that bs = b′ s′ ∈ S and ba = b′ a′ ∈ R.
One denotes an equivalence class containing (s, a) suggestively by as and defines
sa1 + ra2
a1 a2
+
=
s1
s2
ss1
where r ∈ R and s ∈ S are such that rs2 = ss1 ∈ S, and
ra2
a1 a2
·
=
s1 s2
ss1

7

where r ∈ R and s ∈ S are such that rs2 = sa1 ∈ R. One can check that this turn S −1 R
into a ring equipped with an injective ring homomorphism
ϕ : R → S −1 R : r 7→ (1, r)
such that the image under ϕ of every element in S is invertible in S −1 R. We will identify
R with its image ϕ(R) in S −1 R. The ring S −1 R is called the left Ore localisation of R at
S. Finally, note that S −1 R is a flat right R-module and that we can also localise any left
R-module M at S by defining
S −1 M = S −1 R ⊗R M.
If R is a domain and the set of all non-zero elements of R satisfies the left Ore condition
with respect to R, then R is called a left Ore domain.
Returning to our specific setting, note that if G is a torsion-free group such that k[G] is
a domain for a field k, then k[G] is a left Ore domain if G is amenable by [29]. In fact, this
is an if and only if by the appendix of [1] due to Kielak. When exploring these references,
note that one can also consider the ‘right’ Ore condition and all corresponding ‘right’
notions but that for group rings (or more generally rings with an involution), the left Ore
condition and the right Ore condition are equivalent and the corresponding localisations
are isomorphic. For a group G one easily verifies that Z[G] is a left Ore domain if and
only if Q[G] is a left Ore domain. We may therefore conclude that if G is an amenable
group such that Z[G] is a domain, then Z[G] is a left Ore domain and can therefore be
embedded in a ring D = S −1 Z[G] (S = Z[G] \ {0}) such that every non-zero element of
Z[G] is invertible in D. This deals with the setup of Theorem A. In Theorem C however,
G does not need to be amenable. Next, we explain how to proceed in that situation.
Let G be a torsion-free group that satisfies the analytic zero divisor conjecture and
consider the Hilbert spaces
X
X
l2 (G) = {
ag g (ag ∈ C) |
|ag |2 < ∞}
g∈G

g∈G

and
l∞ (G) = {

X

ag g (ag ∈ C) | sup |ag | < ∞}.
g∈G

g∈G

One has the obvious inclusion C[G] ⊆ l2 (G) ⊆ l∞ (G) and the natural ring multiplication
on C[G] extends to a map

X
X
XX
l2 (G) × l2 (G) → l∞ (G) : (α =
ag g, β =
bh h) 7→ αβ =
agh−1 bh g.
g∈G

h∈G

g∈G

h∈H

The group von Neumann algebra N (G) can be defined as (e.g. see [24, Section 8]) as
N (G) = {α ∈ l2 (G) | αβ ∈ l2 (G), ∀β ∈ l2 (G)}

8

and we have an inclusion of rings C[G] ⊆ N (G) such that no non-zero element of C[G] is a
zero-divisor in N (G), by the analytic zero divisor conjecture. Since the set S of non-zero
divisors in N (G) satisfies the left (and right) Ore condition in N (G) (e.g. see [23, Th.
8.22]), one can consider the left (or right) Ore localisation S −1 N (G) which turns out to
be isomorphic to the algebra U(G) of operators affiliated to N (G) (e.g. see [23, Ch. 8] ).
Important for our purposes is that, under the assumptions of Theorem C, we have found
a ring U(G) containing C[G] (and hence Z[G]) such that every non- zero element of C[G]
(and hence Z[G]) is invertible in U(G).
To summarise, if a groups G satisfies the assumptions of either Theorem A or Theorem
C, then the group ring Z[G] can be embedded in a ring D such that every non-zero
element of Z[G] is invertible in D. Our reason for needing such an embedding is contained
in Lemma 3.1 below, which will be used in the proof of Theorem A and C in the next
section.
Lemma 3.1. Let G be a non-trivial amenable group and assume that the group ring Z[G]
can be embedded in a ring D such that
L every non-zero element of Z[G] is invertible in D.
Then for any index set J, the sum j∈J D is an injective Z[G]-module and for all n ≥ 0
Hn (G,

M

D) = 0.

j∈J

Proof. Since every non-zero element of Z[G] isL
invertible in D and G is amenable, it
follows that Z[G] is an Ore domain andL
that
i∈I D is a torsion-free divisible Z[G]module. By [30, Th. 1], this implies that j∈J D is an injective Z[G]-module. From this
L
we immediately conclude that Hn (G, j∈J D) = 0 for all n > 0. Since g − 1 is invertible
L
in D for every g ∈ G we have H0 (G, D) = D G = 0 and hence also H0 (G, j∈J D) = 0. 
Remark 3.2. Let G be a group that contains a copy of the free group on 2 generators
and assume that Z[G] embeds into a ring D such that every non-zero element of Z[G] is
invertible in D (for example, the group ring of any torsion-free one-relator group embeds
into a skew-field by [22]). Then D is a torsion-free divisible Z[G]-module. In particular,
(g − 1)m 6= 0 for any non-trivial g ∈ G and non-zero m ∈ D. But as observed in [26, Ch.
4. Prop. 2.2.] this implies that D is not an injective Z[G]-module. Hence the conclusion
of the lemma above does not hold for groups that contain non-abelian free subgroups.
This also shows that the injective hull I(D) of D viewed as a Z[G]-module cannot be
given the structure of a D-module extending the ring structure of D. Indeed, assume this
was the case and take a non-zero m ∈ I(D). Since I(D) is an essential extension of D
as an Z[G]-module, there exists a non-zero r ∈ Z[G] such that r · m ∈ D. But since r is
invertible in D and the ring structure of D can be extended to an D-module structure on
I(D), we have m = (rr −1) · m = r −1 (r · m) ∈ D. Hence D = I(D) and D is an injective
Z[G]-module, which is a contradiction.

9

4. Proofs of the main results
We are now ready to prove our main results.
proof of Theorem A and C. Let G be a group that satisfies either the assumptions of
Theorem A or Theorem C. Our task is to prove that G of type F P . By Lemma 2.1 it
suffices to show that
lim Hn (G, Fi ) = 0
−→
for any vanishing directed system of free Z[G]-modules {Fi }i∈I . Let {Fi }i∈I be such a
system. As we discussed in the previous section, we can embed the ring Z[G] into a ring
D such that every non-zero element of Z[G] is invertible in D. Consider the short exact
sequence
0 → Z[G] → D → D/Z[G] → 0.
Tensoring with − ⊗Z[G] Fi for every i ∈ I, we obtain a short exact sequence of vanishing
directed systems of left Z[G]-modules
0 → {Fi }i∈I → {D ⊗Z[G] Fi }i∈I → {D ⊗Z[G] Fi /Fi }i∈I → 0.
By considering the associated directed system of long exact cohomology sequences and
taking the direct limit, we obtain the exact sequence
lim Hn−1 (G, D ⊗Z[G] Fi /Fi ) → lim Hn (G, Fi ) → lim Hn (G, D ⊗Z[G] Fi ).
−→
−→
−→
Since G is of type F Pn−1 , it follows from [6, Th. 1.3.]that
lim Hn−1 (G, D ⊗Z[G] Fi /Fi ) = 0.
−→
L
For each i ∈ I, we have D ⊗Z[G] Fi ∼
= j∈J D for some index set J depending on i.
Now let X be an acyclic G-CW complex with amenable stabilisers and without a free
orbit of n-cells. If G is amenable we can choose X = {∗}. Associated to the acyclic
G-CW-complex X, there is a convergent stabiliser spectral sequence (e.g. see [8, Ch.VIII
Section 7])
E1p,q =

Y

σ∈∆p

Hq (Kσ ,

M

D)) =⇒ Hp+q (G,

j∈J

M

D).

j∈J

Here, ∆p is a set of representatives of G-orbits of p-cells of X and Kσ is the stabiliser of
σ. Since every Kσ is amenable, we conclude from Lemma 3.1 that
M
Hq (Kσ ,
D) = 0
j∈J

1
for every q > 0, every p ≥ 0 and every σ ∈ ∆p . Hence, Ep,q
= 0 for all q > 0.

10

Since Kσ is a non-trivial amenable group for every σ ∈ ∆n it also follows from Lemma
3.1 that
1
En,0
=

Y

H0 (Kσ ,

σ∈∆n

M

D) = 0.

j∈J

We conclude that the entireL(p, q)-line with p + q = n of the spectral sequence above is
zero. It follows that Hn (G, j∈J D) = 0 and hence
lim Hn (G, D ⊗Z[G] Fi ) = 0.
−→
n
This implies that lim H (G, Fi ) = 0, which proves Theorems A en C.
−→


proof of Corollary 2. Let Γ be the fundamental group of a graph of groups with cyclic
edge groups and vertex groups that are either subgroups of the additive rational numbers
or solvable Baumslag-Solitar groups. By Bass-Serre theory ([28]), Γ acts on a tree T
(a 1-dimensional acyclic Γ-CW-complex) with cyclic edge (1-cell) stabilisers and vertex
(0-cell) stabilisers that are either subgroups of the additive rational numbers or solvable
Baumslag-Solitar groups. It follows that the stabilisers of 0-cells have cohomological dimension at most two (e.g. see [12]) while the stabilisers of 1-cells have cohomological
dimension at most one. An application of the stabiliser spectral sequence associated to
the action of Γ on T shows that Γ, and hence also every subgroup of Γ, has cohomological
dimension at most 2. Since finitely generated groups are of type F P1 , the corollary follows
from Corollary 1.

proof of Corollary 3. Let H be a non-trivial amenable`subgroup of G such that G commensurates H. Now consider the discrete G-set S = g∈G G/H g and let X denote the
infinite join of S with itself. By construction, there is a cellular and admissible action of
G on X such that the cell-stabilisers are finite intersections of conjugates of H. Since G
commensurates H, we conclude that every cell in X has a non-trivial amenable stabiliser.
Moreover, since taking joins increases the connectivity, X is acyclic. We conclude that X
is an acyclic G-CW complex with amenable stabilisers and no free orbits of n-cells, for
any n. The corollary now follows from Theorem C.

We now turn to the proof of Theorem B.
proof of Theorem B. First consider the case where G is finitely generated. In this case it
follows from Theorem A that G is of type F P . Since amenable groups satisfy the Bass
conjecture by [4], we conclude from [10, Section 4.1] that the ordinary Euler characteristic
of G coincides with the L2 -Euler characteristic of G and is therefore equal to zero (see
also [23]). This implies that H1 (G, C) ∼
= C ⊗Z G/[G, G] 6= 0, proving that G admits a
surjection π onto Z. Since G is of type F P2 , i.e. almost finitely presented, it follows from

11

[7, Theorem A] that G can be realised as an HNN-extension of a finitely generated base
group K with stable letter t mapped to 1 ∈ Z by π. Since G is amenable, it does not
contain non-abelian free subgroups. This forces the HNN-extension to be ascending, by
Britton’s Lemma. Also, by Theorem A, K is of type F P . We claim that cd(K) = 1. To
see this, assume by contradiction cd(K) = 2. Since K must be one ended (since it does
not contain non-abelian free subgroups) this implies that H1 (K, Z[G]) = 0. But by [9,
Th. 0.1], the map α in the Mayer-Vietoris sequence
α

H1 (K, Z[G]) → H2 (G, Z[G]) → H2 (K, Z[G]) −
→ H2 (K, Z[G])
associated to the ascending HNN-extensions G = K∗t is injective. We therefore conclude
that H2 (G, Z[G]) = 0, contradicting the assumption that cd(G) = 2. This proves our
claim, implying that K is free by Stalling’s Theorem [27]. Since K is amenable, this
forces K = Z = hai, proving that G equals a solvable Baumslag-Solitar group
BS(1, m) = ha, t | t−1 at = am i
for some non-zero m ∈ Z.
Now consider the general case. Since every finitely generated group of cohomological
dimension 1 is free by Stallings’ theorem it follows from the above that every finitely
generated subgroup of G either infinite cyclic or a solvable Baumslag-Solitar group. In
particular, G is locally solvable and elementary amenable of finite Hirsch length. Since
torsion-free elementary amenable groups of finite Hirsch length are virtually solvable by
[15], it follows that G must be solvable. By [12, Theorem 5], every solvable group of
cohomological dimension two is isomorphic to either a solvable Baumslag-Solitar group
or a non-cyclic subgroup of the additive rationals.

References
[1] Bartholdi L., Kielak, D. Amenability of groups is characterized by Myhill’s Theorem, arXiv
preprint (2016)
[2] Bartholdi L., Grigorchuk, R., and Sunik, Z., Branch Groups, Handbook of algebra, Vol. 3 (2003),
North-Holland, Amsterdam, 989–1112
[3] Bekka, B., de la Harpe, P., and Valette, A., Kazhdan’s Property (T), New Mathematical Monographs. Cambridge University Press (2008)
[4] Berrick, A. J., Chatterji, I, and, Mislin, G., From acyclic groups to the Bass conjecture for
amenable groups, Math. Ann. Vol. 329(4) (2004), 597–621
[5] Bestvina, M. and Brady, N., Morse theory and finiteness properties of groups, Invent. Math.
129(3) (1997), 445–470
[6] Bieri, R., Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes
[7] Bieri, R., and Strebel, R., Almost finitely presented soluble groups, Comm. Math. Helv. Vol. 53(1)
(1987), 258–278
[8] Brown, K. S., Cohomology of groups, Graduate texts in Mathematics 77, Springer (1982)
[9] Brown, K. S. and Geoghegan R., Cohomology with free coefficients of the fundamental group of
a graph of groups, Comm. Math. Helv. Vol. 60 (1985), 31–45

12

[10] Eckmann, B., Projective and Hilbert modules over group algebras, and finitely dominated spaces,
Comm. Math. Helv. Vol. 71(1) (1996), 453–462
[11] Elek, G., On the analytic zero divisor conjecture of Linnell, Bulletin of the London Mathematical
Society, Vol. 35(2) (2003), 236–238
[12] Gildenhuys, D., Classification of soluble groups of cohomological dimension two, Math. Zeit.,
Vol. 166(1) (1979), 21–25
[13] Grigorchuk, R., and Medynets, K., On algebraic properties of topological full groups, Sbornik:
Mathematics, Vol. 205(6) (2014), 87–108
[14] Hillman, J., A., Elementary amenable groups and 4-manifolds with Euler characteristic0, J. Austral. Math. Soc. Ser. A 50 (1991), 160–170
[15] Hillman, J. A., and Linnell, P. A., Elementary amenable groups of finite Hirsch length are locallyfinite by virtually-solvable, J. Austral. Math. Soc. Ser. A, Vol. 52(2) (1992), 237–241
[16] Juschenko, K., and Monod, N., Cantor systems, piecewise translations and simple amenable
groups, Ann. Math. (2) 178(2) (2013), 775–787
[17] Juschenko, K., Non-elementary amenable subgroups of automata groups, Journal of Topology and
Analysis (to appear)
[18] Kropholler P., Linnell, P., and Lück, W., Groups of small homological dimension and the Atiyah
conjecture, Geometric and Cohomological Methods in Group Theory, LMS Lecture Note Series
358 (2009), 272–277
[19] Kropholler P., Linnell, P., and Moody, J.A, Applications of a New K-Theoretic Theorem to
Soluble Group Rings, Proc. Am. Math. Soc. Vol. 104(3) (1988), 675–684
[20] Lam, T.Y., Lectures on Modules and Rings, Graduate texts in Mathematics 189 (1998), Springer
[21] Leary, I. J. and Saadetoğlu M., The cohomology of Bestvina-Brady groups, Groups, Geometry
and Dynamics 5 (1) (2011), 121-138.
[22] Lewin J. and Lewin T., An embedding of the group algebra of a torsion-free one-relator group in
a field, Journal of Algebra Vol. 52(1) (1978), 39–74
[23] Lück, W., L2-invariants: theory and applications to geometry and K-theory, A Series of Modern
Surveys in Mathematics Vol, 44. Springer-Verlag, Berlin, 2002.
[24] Linnell, P., Analytic versions of the zero divisor conjecture, Geometry and cohomology in group
theory (Durham 1994), London Math. Soc. Lecture Note Series 252, Cambridge University Press
(1998) 209–248.
[25] Olshanskii, A.Y., The Geometry of Defining Relations in Groups, Kluwer Academic Publishers,
(1991)
[26] Roman A.H., Zero Divisors, Group Von Neumann Algebras and Injective Modules, Master thesis,
Virginia Polytechnic Institute and State University (2015)
[27] Stallings J. R., Groups of dimension 1 are locally free, Bull. Amer. Math. Soc. 74 (1968), 361–364
[28] Serre, J.-P., Trees, Springer (1980).
[29] Tamari D., A refined classification of semi-groups leading to generalised polynomial rings with a
generalized degree concept, Proc. ICM vol. 3, Amsterdam (1954) 439–440.
[30] Van de Water, A., A Property of torsion-free modules over left Ore domains, Proc. Am. Math.
Soc. Vol. 25(1) (1970), 199–201
[31] Weibel, C.A., An introduction to homological algebra, Cambridge University Press (1994)
School of Mathematics, Statistics and Applied Mathematics, NUI Galway, Ireland
E-mail address: dieter.degrijse@nuigalway.ie

