# Lamplighters and the bounded cohomology
of Thompson’s group

Nicolas Monod  École Polytechnique Fédérale de Lausanne (EPFL)
CH–1015 Lausanne, Switzerland

###### Abstract.

We prove the vanishing of the bounded cohomology of lamplighter groups for a
wide range of coefficients. This implies the same vanishing for a number of
groups with self-similarity properties, such as Thompson’s group $F$. In
particular, these groups are boundedly acyclic.

Our method is ergodic and applies to “large” transformation groups where the
Mather–Matsumoto–Morita method sometimes fails because not all are acyclic in
the usual sense.

##  1\. Introduction

The initial goal of this note is to prove the following, answering a question
of Grigorchuk [Gri95, p. 131].

######  Theorem 1.

Thompson’s group $F$ is boundedly acyclic.

This statement means that the bounded cohomology
$\mathrm{H}_{\mathrm{b}}^{n}(F)$ vanishes for all $n>0$, where
$\mathrm{H}_{\mathrm{b}}^{n}(-)$ denotes the bounded cohomology (of Gromov
[Gro82] and Johnson [Joh72, §2]) with coefficients in $\mathbf{R}$ viewed as a
trivial module. As for the group $F$, it appeared in many different contexts
[CFP96, CF11] since its 1965 definition by Thompson [Tho65] but the main
outstanding question seems to be whether it is amenable.

One motivation for 1 is that bounded acyclicity is a necessary condition for
amenability, although far from sufficient. In fact, amenability is equivalent
to the vanishing of $\mathrm{H}_{\mathrm{b}}^{n}(-,E)$ with coefficients in
_all_ dual Banach modules $E$, see [Joh72, Thm. 2.5]. In that context, 1 is a
special case of the following.

######  Theorem 2.

The vanishing $\mathrm{H}_{\mathrm{b}}^{n}(F,E)=0$ holds for all $n>0$ and all
separable dual Banach $F$-modules $E$.

This is the first known example of acyclicity for such general coefficients —
except of course amenable groups.

However we caution the reader that this statement does not answer the
amenability question. Indeed, our proof also works for many groups that are
similar to $F$ but known to be non-amenable. For instance, the proof holds
unchanged for all piecewise-projective groups introduced in [Mon13].

In fact, we reduce 2 to a general result on “lamplighter” groups defined as
(restricted) wreath products:

######  Theorem 3.

Let $G$ be any group and consider the wreath product

$W=G\wr\mathbf{Z}=\Big{(}\bigoplus_{\mathbf{Z}}G\Big{)}\rtimes\mathbf{Z}.$

Then $\mathrm{H}_{\mathrm{b}}^{n}(W,E)$ vanishes for all $n>0$ and all
separable dual Banach $W$-modules $E$.

In particular, the case of the trivial module $E=\mathbf{R}$ shows that $W$ is
boundedly acyclic, answering Question 1.8 in [Löh17]. The vanishing statement
of 3 fails if we drop either the separability or the duality assumption.

The connection between this general result for lamplighters and the particular
case of Thompson’s group $F$ comes from the fact that $F$ contains _co-
amenable_ lamplighter subgroups. This condition will be further discussed in
the proof, but for now we point out that this situation is by far not limited
to $F$. Indeed, the following is not much more than a reformulation of 3:

######  Theorem 4.

Let $G$ be a group and $G_{0}<G$ a co-amenable subgroup. Suppose that $G$
contains an element $g$ such that the conjugates of $G_{0}$ by $g^{p}$ and by
$g^{q}$ commute for all $p\neq q$ in $\mathbf{Z}$.

Then $\mathrm{H}_{\mathrm{b}}^{n}(G,E)=0$ holds for all $n>0$ and all
separable dual Banach $G$-modules $E$.

Ignoring for a moment the definition of co-amenability, this result can be
applied under general algebraic conditions that we think of as a form of self-
similarity:

######  Corollary 5.

Let $G$ be a group and $G_{0}<G$ a subgroup with the following two properties:

  1. (i)

every finite subset of $G$ is contained in some $G$-conjugate of $G_{0}$,

  2. (ii)

$G$ contains an element $g$ such that the conjugates of $G_{0}$ by $g^{p}$ and
by $g^{q}$ commute for all $p\neq q$ in $\mathbf{Z}$.

Then $\mathrm{H}_{\mathrm{b}}^{n}(G,E)=0$ holds for all $n>0$ and all
separable dual Banach $G$-modules $E$.

This conclusion holds more generally if (i) is required only for finite
subsets of the derived subgroup $G^{\prime}$ of $G$, or of any (fixed) higher
derived subgroup $G^{(k)}$ of $G$.

Beyond $F$, these conditions are satisfied by many generalisations of this
group, including all non-amenable groups introduced in [Mon13] and their
subgroups studied in [LM16].

We obtain perhaps more intuitive conditions with the following special case,
stated for groups appearing as transformations of some underlying space.
Recall that a transformation is said to be _supported_ on a subset if it is
the identity outside that subset.

######  Corollary 6.

Let $G$ be a group acting faithfully on a set $Z$. Suppose that $Z$ contains a
subset $Z_{0}$ and that $G$ contains an element $g\in G$ with the following
properties:

  1. (i)

every finite subset of $G$ can be conjugated so that all its elements are
supported in $Z_{0}$;

  2. (ii)

$g^{p}(Z_{0})$ is disjoint from $Z_{0}$ for every integer $p\geq 1$.

Then $\mathrm{H}_{\mathrm{b}}^{n}(G,E)=0$ holds for all $n>0$ and all
separable dual Banach $G$-modules $E$. In particular, $G$ is boundedly
acyclic.

Many classical “large” transformation groups satisfy these conditions,
starting with the group of compactly supported homeomorphisms of
$\mathbf{R}^{n}$ which was proved to be boundedly acyclic by Matsumoto–Morita
[MM85]; the latter result was recently widely generalised in [Löh17] and
[FFLM21a].

6 has the following advantage in comparison with the strategy behind the
classical Matsumoto–Morita [MM85] theorem and its generalisations, a strategy
rooted in Mather’s acyclicity theorem for ordinary (co)homology [Mat71].
Namely, the latter relies ultimately on pasting together _infinitely many_
compactly supported transformation (which Berrick describes as occasionally
“difficult to substantiate” [Ber02, 3.1.6]). In our approach, only finitely
many elements need to be pasted together at any one time, according to the
definition of lamplighters, recalling the Aristotelian distinction between
actual and potential infinity.

This makes it possible to apply 6 to less flexible situations, such as
diffeomorphisms or PL homeomorphisms, see [MN21]. By contrast, in ordinary
cohomology, the acyclicity of these less flexible groups is often not true or
unresolved. We refer to [MN21] for applications.

In this context, we mention that Kotschick introduced in [Kot08] a commuting
conjugates condition which resembles one half of our hypothesis in 6 and hence
holds more generally. He used it to prove the vanishing of the stable
commutator length [Cal09], which according to Bavard’s result [Bav91] follows
from the vanishing of $\mathrm{H}_{\mathrm{b}}^{2}(-,\mathbf{R})$. There is no
reason, however, that Kotschick’s groups should all be boundedly acyclic or
satisfy the vanishing with coefficients. Similar comments hold for the very
recent “commuting conjugates” criterion for the vanishing of
$\mathrm{H}_{\mathrm{b}}^{2}(-,\mathbf{R})$ given in [FFL21].

######  Acknowledgements.

I am indebted to S. Nariman for rekindling my interest in the question of
bounded acyclicity. I am grateful to F. Fournier-Facio, C. Löh and M.
Moraschini for showing me their preprint [FFLM21a], for referring me to
Grigorchuk’s question and for many comments.

##  2\. Lamplighters

We shall first work towards the proof of 3 for the case of a _countable_ group
$G$. This restriction will be lifted in Section 2.D.

###  2.A. Ergodicity

In order to handle non-trivial coefficients, we recall the notion of
_ergodicity with coefficients_ introduced with M. Burger in [BM02]. The reader
only interested in bounded acyclicity can skip this discussion.

Consider a group $G$ with a non-singular action on a standard probability
space $X$. Recall that usual ergodicity is equivalent to the statement that
every $G$-invariant measurable function (class) $f\colon X\to\mathbf{R}$ is
(essentially) constant. There is no difference if instead $f\colon X\to E$
ranges in a separable Banach space, or indeed any polish space. However, if
$E$ is endowed with a non-trivial $G$-representation, then the requirement
that every $G$-_equivariant_ measurable function class $f\colon X\to E$ be
essentially constant is much stronger. This is called ergodicity with
coefficients in $E$. A standard fact is that ergodicity with coefficients in
_separable_ Banach modules follows from the ergodicity of the diagonal action
on $X^{2}$. It is important here to recall that Banach modules are always
assumed to be endowed with an _isometric_ $G$-representation. The proof
consists in observing that $f\colon X\to E$ must satisfy that
$\|f(x)-f(x^{\prime})\|$ is essentially constant, and that this constant must
be zero because $E$ is second countable.

One of the earliest examples of an ergodic action is the _Bernoulli shift_ ,
defined as follows. Let $X$ be a standard probability space and consider the
countable power $Y=X^{\mathbf{Z}}$. Then the shift of coordinates is an
ergodic $\mathbf{Z}$-action on $Y$; this is essentially the same statement as
Kolmogorov’s zero-one law. It follows that the diagonal action on any power
$Y^{d}$ is also ergodic, because $Y^{d}$ can be identified with
$(X^{d})^{\mathbf{Z}}$ in a $\mathbf{Z}$-equivariant manner. Considering $2d$
instead of $d$, we can upgrade this to ergodicity with separable coefficients.

###  2.B. Amenable actions and co-amenable subgroups

In the proof of 3, we shall use the terminology of _amenable actions in
Zimmer’s sense_ in the context of a countable group with a non-singular action
on a standard probability space. We refer to [Zim78] or to [Zim84, §4] for the
definition and recall the following basic facts.

######  Lemma 7.

(i) Consider Zimmer-amenable actions of $G_{i}$ on $X_{i}$ for $i=1,2$. Then
the product action of $G_{1}\times G_{2}$ on $X_{1}\times X_{2}$ is Zimmer-
amenable.

(ii) Suppose that $G$ is the union of an increasing sequence of subgroups
$G_{n}<G$, $n\in\mathbf{N}$. A non-singular action of $G$ is Zimmer-amenable
if the corresponding action of every $G_{n}$ is Zimmer-amenable.

###### Proof.

Both statements follow readily from Zimmer’s original definition. They also
both follow from the equivalent characterisation given in [AEG94, Thm. A(v)].
∎

######  Corollary 8.

For each integer $n$, let $G_{n}$ be a countable group with a Zimmer-amenable
non-singular action on a standard probability space $X_{n}$. Then the action
of the restricted product $\bigoplus_{n}G_{n}$ on the (unrestricted) product
$\prod_{n}X_{n}$ is a Zimmer-amenable non-singular action on a standard
probability space.

###### Proof.

Viewing $\bigoplus_{n}G_{n}$ as an increasing union of finite products, this
follows from combining the two statements of 7. Note that the action is non-
singular since any given group element acts on only finitely many coordinates.
∎

We now observe that there is a nice interplay between Zimmer-amenability and
Eymard’s notion of co-amenability [Eym72]. Recall that a subgroup $H<G$ is
_co-amenable in $G$_ if there is a $G$-invariant mean on $G/H$; another
equivalent condition is recalled in the proof below.

######  Proposition 9.

Let $G$ be a countable group with a non-singular action on a standard
probability space $X$. Let $H<G$ be a co-amenable subgroup. If the
corresponding $H$-action on $X$ is Zimmer-amenable, then so is the $G$-action.

###### Proof.

Recall that a group $G$ is amenable if and only if every convex compact
$G$-set $K\neq\varnothing$ (in a Hausdorff locally convex space) has a
$G$-fixed point; we can moreover assume that the ambient space is the dual of
a Banach $G$-module in the weak-* topology, and for countable $G$ it suffices
to consider duals of a separable spaces.

More generally, a subgroup $H<G$ is co-amenable if this condition holds for
the subclass of those G-sets $K$ which have an $H$-fixed point.

Finally, Zimmer’s definition of the amenability of the $G$-action on $X$ is
the fixed-point property for the following specific class of $G$-sets $K$.
Start with a measurable _field_ of convex compact $K_{x}\neq\varnothing$
($x\in X$), but with a cocycle action over $G\times X$ instead of a
$G$-action. Then the requirement of Zimmer’s definition is a $G$-fixed point
in the convex compact $G$-set $K=L^{\infty}(X,K_{\bullet})$ of measurable
sections. Since the $G$-action on $K$ combines the $G$-action on $X$ with the
cocycle action on $\\{K_{x}\\}$, a section is $G$-fixed exactly when it is
cocycle-equivariant as a map; compare [Zim84, 4.3.1].

Combining these reformulations of the definitions yields the statement of the
proposition. ∎

Since we just recalled the definition(s) of co-amenability, we can see how
this notion is relevant to condition (i) in 5:

######  Proposition 10.

Let $G$ be any group and $G_{0}<G$ a subgroup.

If every finite subset of $G$ is contained in some conjugate of $G_{0}$, then
$G_{0}$ is co-amenable in $G$.

This holds more generally if every finite subset of a fixed co-amenable
subgroup $G_{1}<G$ is contained in some $G$-conjugate of $G_{0}$.

The more general form above is relevant to the additional statement in 5 by
setting $G_{1}=G^{(k)}$; indeed $G^{(k)}$ is co-amenable in $G$ since it is
normal with soluble quotient.

###### Proof of 10.

Let $K$ be a convex compact $G$-set containing some $G_{0}$-fixed point $k$.
We need to prove that $K$ has a $G$-fixed point, but it suffices to show that
it has a $G_{1}$-fixed point since the latter is co-amenable in $G$. Given any
finite subset $F$ of $G_{1}$, let $g_{F}\in G$ be an element conjugating $F$
into $G_{0}$. Then $g_{F}k$ is fixed by the group generated by $F$. Consider
$g_{F}k$ as a net indexed by the directed set of all finite subsets $F$ of
$G_{1}$. Then any accumulation point of this net in the compact space $K$ will
be fixed by $G_{1}$. ∎

The connection between co-amenability and bounded cohomology is the following
basic fact. A proof can be found e.g. in [Mon01, Prop. 8.6.6] and actually it
provides a characterisation of co-amenability, see [MP03, Prop. 3]. (There is
a countability assumption in [Mon01, Prop. 8.6.6] but it is not needed nor
used.)

######  Proposition 11.

Let $G$ be a group, $G_{0}<G$ a co-amenable subgroup and $E$ a dual Banach
$G$-module. Then the restriction map

$\mathrm{H}_{\mathrm{b}}^{\bullet}(G,E)\longrightarrow\mathrm{H}_{\mathrm{b}}^{\bullet}(G_{0},E)$

is injective.∎

###  2.C. Vanishing for countable wreath products

Let $G$ be a countable group, $W=G\wr\mathbf{Z}$ the (restricted) wreath
product and $E$ a separable dual $W$-module.

Let further $X$ be any standard probability space with a non-singular
$G$-action which is amenable in Zimmer’s sense. One can for instance simply
take $X$ to be $G$ itself, endowed with any distribution of full support. The
countable power $Y=X^{\mathbf{Z}}$ is a standard probability space with a non-
singular action of $\bigoplus_{n\in\mathbf{Z}}G$ which is Zimmer-amenable by
8. We let $\mathbf{Z}$ act on $Y$ by shifting the coordinates; that is, $Y$ is
the $X$-based Bernoulli shift. Combining the two actions, we have thus endowed
$Y$ with an action of the (restricted) wreath product $W=G\wr\mathbf{Z}$.

The subgroup $\bigoplus_{n\in\mathbf{Z}}G$ is co-amenable in $W$ since it is a
normal subgroup with amenable quotient. Therefore, the above discussion allows
us to apply 9 and conclude that the $W$-action on $Y$ is Zimmer-amenable.

According to [BM02, Thm. 2] or to [Mon01, Thm. 7.5.3], the bounded cohomology
of $W$ with coefficient in $E$ is realised by the complex of $W$-equivariant
measurable bounded function classes

$0\longrightarrow L^{\infty}(Y,E)^{W}\longrightarrow
L^{\infty}(Y^{2},E)^{W}\longrightarrow
L^{\infty}(Y^{3},E)^{W}\longrightarrow\cdots$

with the usual “simplicial” Alexander–Kolmogorov–Spanier differentials. (The
general references above specify that measurability is in the weak-* sense,
but this is irrelevant here, see [Mon01, Lem. 3.3.3].)

The discussion of Section 2.A shows that the $\mathbf{Z}$-equivariant elements
of $L^{\infty}(Y^{d},E)$ are essentially constant, and hence so is every
element of $L^{\infty}(Y^{d},E)^{W}$. It follows that the latter space
consists of all essentially constant maps ranging in $E^{W}$. Since the
simplicial differentials are alternating sums of the map omitting each
variable, we conclude that the above complex of $W$-equivariant maps is
acyclic except possibly in degree zero, where its cohomology is $E^{W}$. This
completes the proof of 3 when $G$ is countable.

###  2.D. To $\aleph_{0}$ and beyond

We imposed a countability assumption on our groups in order to be able to
apply standard ergodic methods straight out of the shipping box. There does
not appear to be a deeper reason that countability should be needed. In any
case, the following general principle will allow us to reduce 3 to the
countable case.

######  Proposition 12.

Let $G$ be any group and $E$ a separable dual Banach $G$-module. Suppose that
every countable subset of $G$ is contained in a subgroup $G_{1}<G$ such that
$\mathrm{H}_{\mathrm{b}}^{n}(G_{1},E)$ vanishes for all $n>0$.

Then $\mathrm{H}_{\mathrm{b}}^{n}(G,E)$ vanishes for all $n>0$.

Since the classical examples of boundedly acyclic groups were precisely large,
uncountable, groups, we single out the following particular case of 12.

######  Corollary 13.

Let $G$ be any group. Suppose that every countable subset of $G$ is contained
in some boundedly acyclic subgroup of $G$.

Then $G$ is boundedly acyclic.∎

###### Proof of 12.

Consider a group $J$ and a separable Banach $J$-module $E$ which is the dual
of some Banach $J$-module $F$. The key claim is that
$\mathrm{H}_{\mathrm{b}}^{n}(J,E)$ vanishes for all $n>0$ if and only if the
$\ell^{1}$-homology $\mathrm{H}^{\ell^{1}}_{n}(J,F)$ vanishes for all $n>0$.
This fact then implies the proposition when we apply it to both $G$ and
$G_{1}$, viewing $E$ also as a dual Banach $G_{1}$-module.

The claim is established as [MM85, Cor. 2.4(iii)] in the special case of $E$
trivial, which is sufficient for 13.

In the general case, we recall first that $\mathrm{H}_{\mathrm{b}}^{n}(J,E)$
vanishes for all $n>0$ if and only if the following two conditions hold:
first, $\mathrm{H}^{\ell^{1}}_{n}(J,F)$ vanishes for all $n>0$; second,
$\mathrm{H}^{\ell^{1}}_{0}(J,F)$ is Hausdorff. This equivalence relies on the
closed range theorem and was already established by Johnson [Joh72, Cor. 1.3],
see also [MM85, Cor. 2.4(i),(ii)]. Moreover, the second condition is
equivalent to $\mathrm{H}_{\mathrm{b}}^{1}(J,E)$ being Hausdorff, see [MM85,
Thm. 2.3]

In conclusion, it certainly suffices to prove
$\mathrm{H}_{\mathrm{b}}^{1}(J,E)=0$. This amounts to showing that every
affine isometric action with bounded orbits on a separable dual Banach space
admits a fixed point. That statement is a variant of the Ryll-Nardzewski
theorem. Specifically, it is the case (c) in Bourbaki’s account, Appendix 3 to
part IV of [Bou87]. Warning: the English translation incorrectly requires
first countability for the norm of $E$, which is both empty and insufficient,
whereas the proof and the French original correctly use second countability,
which here is just the separability of $E$ that we assumed. ∎

###### End of proof of 3.

Let $G$ by an arbitrary group and $S\subseteq G\wr\mathbf{Z}$ a countable
subset of the wreath product. Then there is a countable subgroup $G_{1}$ of
$G$ such that $G_{1}\wr\mathbf{Z}$ contains $S$. In consequence, 12 allows us
to reduce the general case to the case of wreath products of countable groups.
∎

##  3\. Self-similar situations

It is shaped, sir, like itself; and it is as broad as it hath breadth: it is
just so high as it is, and moves with its own organs

Shakespeare, _Antony and Cleopatra_ , Act 2, Scene 7

We now consider the situation where a group contains a suitable supply of
copies of a given co-amenable subgroup; in particular the case of Corollaries
5 and 6 where the replicating subgroup also progressively swallows the entire
ambient group.

###### Proof of 4..

Let $G$ be a group, $G_{0}<G$ a co-amenable subgroup and suppose that $G$
contains an element $g$ such that the conjugates of $G_{0}$ by $g^{p}$ and by
$g^{q}$ commute for all $p\neq q$ in $\mathbf{Z}$. Let further $E$ be a
separable dual Banach $G$-module.

Let $W_{1}<G$ be the subgroup generated by $G_{0}$ and $g$. Then $W_{1}$ is
co-amenable in $G$ since it contains $G_{0}$. Therefore, by 11, it suffices to
show that $\mathrm{H}_{\mathrm{b}}^{n}(W_{1},E)$ vanishes for all $n>0$. The
commutativity assumption implies that there is a natural map from the wreath
product $W=G_{0}\wr\mathbf{Z}$ onto $W_{1}$ mapping $1\in\mathbf{Z}$ to $g$.
This turns $E$ into a $W$-module and 3 implies the vanishing of
$\mathrm{H}_{\mathrm{b}}^{n}(W,E)$. It only remains to justify that the
inflation

$\mathrm{H}_{\mathrm{b}}^{n}(W_{1},E)\longrightarrow\mathrm{H}_{\mathrm{b}}^{n}(W,E)$

is injective. By construction, the kernel of the projection $W\to W_{1}$ is
metabelian and hence amenable. This implies that the inflation is an
isomorphism (see [Nos91, Thm. 1] or [Mon01, Rem. 8.5.4]) and thus 4 follows. ∎

###### Proof of 5.

Let $G$ be a group, $E$ a separable dual Banach $G$-module and $G_{0}<G$ a
subgroup. To prove the general case of 5, we can assume the following.

  1. (i)

Every finite subset of a (fixed) co-amenable subgroup $G_{1}<G$ is contained
in some $G$-conjugate of $G_{0}$.

  2. (ii)

$G$ contains an element $g$ such that the conjugates of $G_{0}$ by $g^{p}$ and
by $g^{q}$ commute for all $p\neq q$ in $\mathbf{Z}$.

In view of 10, the first condition implies that $G_{0}$ is co-amenable in $G$.
Therefore, we are in a position to apply 4 and hence 5 is established. ∎

###### Proof of 6.

We are given a group $G$ acting faithfully on a set $Z$, an element $g\in G$
and a subset $Z_{0}\subseteq Z$ such that:

  1. (i)

every finite subset of $G$ can be conjugated so that all its elements are
supported in $Z_{0}$;

  2. (ii)

$g^{p}(Z_{0})$ is disjoint from $Z_{0}$ for every integer $p\geq 1$.

Define $G_{0}<G$ to be the subgroup consisting of all elements of $G$
supported in $Z_{0}$. The first assumption implies that every finite subset of
$G$ is contained in some conjugate of $G_{0}$. Next, note that the sets
$g^{p}(Z_{0})$ and $g^{q}(Z_{0})$ are disjoint for all distinct
$p,q\in\mathbf{Z}$. Since $G$ acts faithfully, this implies that the
conjugates of $G_{0}$ by $g^{p}$ and by $g^{q}$ commute. Thus 6 follows indeed
from 5. ∎

We can now apply this result to the case of Thompson’s group $F$. Since this
group has a number of very different descriptions linking it to interesting
objects in homotopy, algebra and combinatorics, we should specify which
description of $F$ we work with. We consider $F$ to be the group of piecewise
affine homeomorphisms of $[0,1]$ with dyadic breakpoints and slopes in
$2^{\mathbf{Z}}$. We refer to [CFP96] for background.

###### Proof of 2 and hence also of 1.

We work with the derived subgroup $F^{\prime}$ of $F$, which is sufficient by
11. Choose a non-trivial element $g\in F^{\prime}$ and choose a dyadic point
$x_{0}\in(0,1)$ not fixed by $g$. Let $Z$ be the $F^{\prime}$-orbit of $x_{0}$
(which happens to consist of all dyadic points of $(0,1)$). Define $Z_{0}$ to
be the open interval determined by $x_{0}$ and $g(x_{0})$, which is non-empty
by construction.

The first condition of 6 is satisfied by the transitivity properties of the
$F^{\prime}$-action: any interval stricly contained in $(0,1)$ can be shrunk
into $Z_{0}$ (compare e.g. the proof of Cor. 2.3 in [CM09]). The second
condition follows from the fact that $g$ preserves the order on $(0,1)$. In
conclusion, we are in a situation to appeal to 6 and thus complete the proof
of 2 and hence also of 1. ∎

(The reader might notice that this reasoning, when brought all the way back to
the underlying wreath product subgroup of $F$, is an improvement of our
comments in Section 6.C of [Mon10].)

The above argument holds for many similar groups since it only relies on the
abstract statement of 6; as mentioned in the introduction, this includes all
piecewise-projective groups of homeomorphisms of the line that have
sufficiently transitive orbits.

##  4\. Further comments

###  4.A. More acyclicity

First, we should recall that many examples of boundedly acyclic groups (with
trivial coefficients) have been discovered, starting with the theorem of
Matsumoto–Morita [MM85]. Recent examples include, among others, [Löh17],
[FFLM21a], [FFLM21b], [MN21]. A very nice general criterion, but for degree
two only, is given in [FFL21].

Furthermore, boundedly acyclic groups can be used as a tool in results aiming
to determine _non-trivial_ bounded cohomology of larger groups. This has
recently led to the complete computation of the bounded cohomology of some
groups that are not boundedly acyclic [MN21]. These methods, or the methods of
[FFLM21a, §6], can now leverage 1 above to establish that the bounded
cohomology of Thompson’s circle group $T$ is generated by the bounded Euler
class, in perfect analogy to the result of [MN21] about the entire group of
(orientation-preserving) homeomorphisms of the circle. By contrast, the usual
cohomology of $T$ (and of $F$) is richer and completely described in [GS87].

###  4.B. More lamplighters

In the proof of the vanishing for wreath products, the only properties of
$\mathbf{Z}$ that we used were that it is infinite and amenable. Therefore,
the same results holds more generally for all (restricted) wreath products
$G\wr\Gamma$ as long as $\Gamma$ is infinite amenable.

A closer examination of the proof also shows that it holds for suitable
permutational wreath products where $\mathbf{Z}$ is replaced by an amenable
group with a permutation action having only infinite orbits; the ergodicity of
the corresponding generalised Bernoulli shifts is recorded e.g. in [KT08,
Prop. 2.1].

These generalisations can in turn be used to extend the statements of 4 and of
5. Specifically, instead of a single element $g$ and the corresponding
commuting conjugates $G_{0}^{g^{p}}$, it suffices to assume there is some
infinite amenable subgroup $\Gamma<G$ such that the $\Gamma$-conjugates of
$G_{0}$ commute pairwise. We can proceed similarly to generalise 6.

###  4.C. No more coefficients

As recalled in the introduction, 3 does not hold without the separability
assumption on the dual module $E$. Indeed, in that case the vanishing of
$\mathrm{H}_{\mathrm{b}}^{n}(-,E)$ for all $n>0$ is equivalent to amenability.
(3 does hold for some very specific non-separable modules such as the _semi-
separable_ case introduced in [Mon10], but only because they can be reduced to
the separable case.)

A very concrete example, not relying on the huge coefficient module witnessing
non-amenability in general, is as follows. Choose a group $G$ with
$\mathrm{H}_{\mathrm{b}}^{2}(G)\neq 0$, for instance a free group of rank two.
Then, by inflation,
$\mathrm{H}_{\mathrm{b}}^{2}\big{(}\bigoplus_{\mathbf{Z}}G\big{)}$ is also
non-zero. It follows by cohomological induction [Mon01, §10.1] that

$\mathrm{H}_{\mathrm{b}}^{2}\big{(}G\wr\mathbf{Z},\ell^{\infty}(\mathbf{Z})\big{)}\
\neq 0,$

where $G\wr\mathbf{Z}$ acts on $\ell^{\infty}(\mathbf{Z})$ by translation via
the quotient morphism $G\wr\mathbf{Z}\to\mathbf{Z}$. The only circumstance
preventing a contradiction with 3 is that the dual Banach module
$\ell^{\infty}(\mathbf{Z})$ is not separable.

We mention here that Grigorchuk asked about the vanishing of
$\mathrm{H}_{\mathrm{b}}^{n}(F,E)$ for all $n\geq 2$ and all dual $E$, see
Problem 3.19 in [Gri95]. This is a priori a weakening of amenability, first
considered by Johnson in §10.10 of [Joh72]. However, it was shown in [Mon06,
Cor. 5.10] that this condition is in fact also equivalent to amenability; this
result relies on the Gaboriau–Lyons theorem [GL09].

The condition that $E$ be _dual_ cannot be removed either (even when keeping
it separable). Indeed it is well-known that the vanishing of
$\mathrm{H}_{\mathrm{b}}^{1}(G,E)$ for all Banach modules characterises
_finite_ groups $G$. This follows by applying the cohomological long exact
sequence (for bounded cohomology) to the submodule inclusion
$\ell^{1}_{0}(G)\to\ell^{1}(G)$, where $\ell^{1}_{0}(G)$ denotes the summable
functions with vanishing sum. This module is separable when $G$ is countable.

###  4.D. Amenability vs. Ergodicity

It is a remarkable fact (in the author’s opinion) that _every_ group admits a
Zimmer-amenable space $X$ which is doubly ergodic with separable coefficients.
An early proof is found in [BM99, BM02] and the most luminous argument is in
[Kaĭ03]. This cannot be extended to higher ergodicity in general precisely
because non-trivial bounded cohomology provides an obstruction.

This fact, together with the observation that the amenability of $F$ is
equivalent to the Zimmer-amenability of the $T$-action on the circle, has
prompted us to ask whether non-amenable groups can have Zimmer-amenable
actions that are multiply ergodic far beyond two factors, see Problem H in
[Mon06]. The construction of Section 2.C shows that this is indeed possible.

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