2.1 Bounded cohomology with dual coefficients

Every group in this paper is assumed to be discrete. In this section, we recall the basic definitions of bounded cohomology of groups [Reference FrigerioFri17, Reference MonodMon01].

Let $\Gamma$ be a group. We say that E is a Banach $\Gamma$ -module if E is a complete normed $ {\mathbb{R}}$ -module admitting a $\Gamma$ -action by linear isometries. We refer to E as a trivial module when the $\Gamma$ -action is the trivial action. In this paper, the Banach $\Gamma$ -module ${\mathbb{R}}$ will be always assumed to be trivial, unless specified otherwise.

Given a group $\Gamma$ and a Banach $\Gamma$ -module E, we can then define the cochain complex of bounded functions,

\begin{align*}C_b^\bullet(\Gamma; E) := \{f \colon \Gamma^{\bullet + 1} \to E \, \mid \, \| f \|_\infty := \sup_{\gamma_0, \ldots, \gamma_\bullet} \|f(\gamma_0, \ldots, \gamma_\bullet)\|_E< +\infty\} , \end{align*}

endowed with the simplicial coboundary operator

\begin{align*}\delta^\bullet \colon C_b^\bullet(\Gamma; E) \to C_b^{\bullet+1}(\Gamma; E) \end{align*}

$$\delta^\bullet f(g_0,\ldots,g_{\bullet+1})=\sum_{0\leq i\leq \bullet+1} (-1)^i f(g_0,\ldots,\hat{g_i},\ldots,g_{\bullet+1})$$

that sends bounded functions to bounded functions (we will typically simply denote it by $\delta$ when the degree is clear from the context). By construction, $\Gamma$ acts on $C_b^\bullet(\Gamma; E)$ as follows. For every $f \in C_b^\bullet(\Gamma; E)$ ,

(1) \begin{equation}(\gamma \cdot f)(\gamma_0, \ldots, \gamma_\bullet) = \gamma f(\gamma^{-1} \gamma_0, \ldots, \gamma^{-1}\gamma_\bullet)\end{equation}

for all $\gamma, \gamma_0, \ldots, \gamma_\bullet \in \Gamma$ . We denote by $C_b^\bullet(\Gamma; E)^\Gamma$ the subspace of $\Gamma$ -invariant bounded functions. It is immediate to check that the coboundary operator is $\Gamma$ -equivariant, so that $(C_b^\bullet(\Gamma; E)^\Gamma, \delta^\bullet)$ is also a cochain complex. We denote by $Z^\bullet_b(\Gamma; E)$ its cocycles and by $B^\bullet_b(\Gamma; E)$ its coboundaries.

Definition 2.1 (Bounded cohomology). We define the bounded cohomology of $\Gamma$ with coefficients E as the cohomology of the cochain complex $(C_b^\bullet(\Gamma; E)^\Gamma, \delta^\bullet)$ :

\begin{align*} H_b^\bullet(\Gamma; E) := H^\bullet(C_b^\bullet(\Gamma; E)^\Gamma, \delta^\bullet) = Z^\bullet_b(\Gamma; E)/B^\bullet_b(\Gamma; E). \end{align*}

Remark 2.2 (Comparison map). Since the cochain complex of $\Gamma$ -invariant bounded functions sits inside the cochain complex of $\Gamma$ -invariant (possibly unbounded) functions, there exists a canonical map, called a comparison map,

\begin{align*}\textrm{comp}^{\bullet} \colon H_b^\bullet(\Gamma; E) \to H^\bullet(\Gamma; E), \end{align*}

induced by the inclusion.

Following Monod’s notation [Reference MonodMon01, § 11.1], given a group $\Gamma$ , we denote by $\mathfrak{X}$ a class of Banach $\Gamma$ -modules. Throughout the paper, we will exclusively consider the following three classes.

1. (1) We denote by $\mathfrak{X}^{\text{dual}}$ the class of all dual Banach $\Gamma$ -modules. We recall that a Banach $\Gamma$ -module lies in $\mathfrak{X}^{\text{dual}}$ if it is isometrically $\Gamma$ -isomorphic to the dual of some Banach $\Gamma$ -module.

2. (2) We denote by $\mathfrak{X}^{\text{sep}}$ the class of all separable dual Banach $\Gamma$ -modules.

3. (3) We denote by $\mathfrak{X}^{\{{\mathbb{R}}\}}$ the class consisting only of the Banach $\Gamma$ -module $ {\mathbb{R}}$ endowed with the trivial $\Gamma$ -action.

By definition, we have the following inclusions:

(*) \begin{equation}\mathfrak{X}^{\{{\mathbb{R}}\}} \subset \mathfrak{X}^{\text{sep}} \subset \mathfrak{X}^{\text{dual}}.\end{equation}

The chain of inclusions given in Equation (*) shows that the classes of $\mathfrak{X}$ -boundedly acyclic (BAc) groups satisfy the following:

\begin{align*}\mathfrak{X}^{\text{dual}}\mbox{-}\text{BAc} \subseteq \mathfrak{X}^{\text{sep}}\mbox{-}\text{BAc} \subseteq \mathfrak{X}^{\{ {\mathbb{R}}\}}\mbox{-}\text{BAc}. \end{align*}

However, there exists a group that is $\mathfrak{X}^{\{{\mathbb{R}}\}}$ -boundedly 2-acyclic but is not $\mathfrak{X}^{\text{sep}}$ -boundedly 2-acyclic, namely $\Gamma := \overline{T} \rtimes \mathbb{Z}/2$ , where $\overline{T}$ is the lift of Thompson’s group to the real line, and the order-2 automorphism is given by the change of orientation. Indeed, it follows by a combination of two results, [Reference Fournier-Facio and WadeFFW23, Example 6.12] and [Reference MonodMon01, Example 12.4.3], that $H^2_b(\Gamma; {\mathbb{R}}) = 0$ . However, by Eckmann–Shapiro induction [Reference MonodMon01, Proposition 10.1.3], there exists an isomorphism $H^2_b(\Gamma; \ell^\infty(\Gamma / \overline{T}, {\mathbb{R}})) \cong H^2_b(\overline{T}; {\mathbb{R}}) \neq 0$ (via the rotation quasimorphism [Reference CalegariCal09]); but $\ell^\infty(\Gamma / \overline{T}, {\mathbb{R}})$ is two-dimensional, and thus dual (reflexive) and separable.

Here is a list of examples of $\mathfrak{X}$ -boundedly acyclic groups.

1. (1) All amenable groups are $\mathfrak{X}^{\text{dual}}$ -boundedly acyclic [Reference JohnsonJoh72].

2. (2) Thompson’s group F is $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic [Reference MonodMon22].

3. (3) Let $\Lambda$ be a group. All wreath products

   \begin{align*} \Lambda \wr {\mathbb{Z}} = (\bigoplus_{{\mathbb{Z}}} \Lambda) \rtimes {\mathbb{Z}} \end{align*}
   are $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic [Reference MonodMon22].

4. (4) Every countable group embeds into a $\mathfrak{X}^{\{{\mathbb{R}}\}}$ -boundedly acyclic group, and there exists a finitely presented $\mathfrak{X}^{\{{\mathbb{R}}\}}$ -boundedly acyclic group that contains all finitely presented groups [Reference Fournier-Facio, Löh and MoraschiniFFLM24].

5. (5) All binate groups [Reference BerrickBer89] are $\mathfrak{X}^{\{{\mathbb{R}}\}}$ -boundedly acyclic [Reference Matsumoto and MoritaMM85, Reference LöhLöh17, Reference Fournier-Facio, Löh and MoraschiniFFLM23].

6. (6) The groups of homeomorphisms and diffeomorphisms of compact support of manifolds of the form ${\mathbb{R}}^n \times M$ , with M closed and $n\geq 1$ , are $\mathfrak{X}^{\{{\mathbb{R}}\}}$ -boundedly acyclic [Reference Monod and NarimanMN23].

We end this section with Gromov’s foundational mapping theorem, which will be useful in some arguments throughout the text.

In particular, given a class $\mathfrak{X} \subseteq \mathfrak{X}^{\text{dual}}$ of Banach $\Lambda$ -modules, if $\Gamma$ is $\mathfrak{X}$ -boundedly n-acyclic, then $\Lambda$ is also $\mathfrak{X}$ -boundedly n-acyclic.

Recall that a subgroup $\Lambda$ of a group $\Gamma$ is co-amenable in $\Gamma$ if $\ell^{\infty}(\Gamma/\Lambda)$ admits a $\Gamma$ -invariant mean.

In particular, given a class $\mathfrak{X} \subseteq \mathfrak{X}^{\text{dual}}$ of Banach $\Gamma$ -modules, if $\Lambda$ is $\mathfrak{X}$ -boundedly n-acyclic, then $\Gamma$ is also $\mathfrak{X}$ -boundedly n-acyclic.

2.2 Highly ergodic Zimmer-amenable actions

We recall in this section how to compute bounded cohomology using highly ergodic Zimmer-amenable actions. Recall that a Borel space is standard if it is Borel isomorphic to the Borel space associated to a Polish space. Given a non-singular (i.e. measure class preserving) action of a group $\Gamma$ on a standard Borel measure space S, and a Banach $\Gamma$ -module E, there is an induced action of $\Gamma$ on $L^\infty(S, E)$ , defined in the same way as the action of $\Gamma$ on $C^n_b(\Gamma; E)$ (see Equation (1)).

Definition 2.9 (Zimmer-amenable space) Let $\Gamma$ be a group. Let S be a standard Borel probability space, equipped with a non-singular action of $\Gamma$ . A conditional expectation

\begin{align*}m \colon L^\infty(\Gamma \times S, {\mathbb{R}}) \to L^\infty(S, {\mathbb{R}}) \end{align*}

is a norm-1 linear map such that:

1. (1) $m(\mathbb{1}_{\Gamma \times S}) = \mathbb{1}_S$ ; and

2. (2) for all $f \in L^\infty(\Gamma \times S, {\mathbb{R}})$ andvery measurable subset $A \subset S$ , it holds that $m(f \cdot \mathbb{1}_{\Gamma \times A}) = m(f) \cdot \mathbb{1}_A$ .

We say that S is a Zimmer-amenable $\Gamma$ -space if there exists a conditional expectation $m \colon L^\infty(\Gamma \times S) \to L^\infty(S)$ that is, moreover, $\Gamma$ -equivariant (on $\Gamma\times S$ , we consider the diagonal $\Gamma$ -action).

The following proposition provides useful examples of Zimmer-amenable spaces that will be of crucial use for us [Reference MonodMon22, § 2.3, Corollary 8 and Proposition 9].

Zimmer-amenable spaces play a fundamental role in the theory of bounded cohomology.

Theorem 2.11 (Bounded cohomology and Zimmer-amenable spaces [Reference MonodMon01, Theorem 7.5.3]). Let $\Gamma$ be a countable group, let $E \in \mathfrak{X}^{\text{sep}}$ be a Banach $\Gamma$ -module, and let S be a Zimmer-amenable $\Gamma$ -space. Then, the complex

\begin{align*} 0 \to L^\infty(S, E)^{\Gamma} \to L^\infty(S^2, E)^{\Gamma} \to L^\infty(S^3, E)^{\Gamma} \to \cdots\!, \end{align*}

endowed with the simplicial coboundary operator, isometrically computes the bounded cohomology of $\Gamma$ with coefficients in E.

This comes in particularly handy when $L^\infty(S^{n+1}, E)^{\Gamma}$ is small. Recall the following definition.

Before moving on, we introduce the following definitions.

Definition 2.14 (The generalized Bernoulli shift). Let $\Gamma$ be a countable group acting on a countable set X. Let Y be a standard Borel probability space and assume that the measure does not concentrate on a single point. We can then define a new standard Borel probability space $Y^X = \prod_{x \in X} Y$ . The generalized Bernoulli shift is the dynamical system given by the $\Gamma$ -action of $Y^X$ , defined as follows:

\begin{align*}\gamma \cdot(y_x)_{x \in X} := (y_{\gamma^{-1} \cdot x})_{x \in X}, \end{align*}

for all $\gamma \in \Gamma$ and $(y_x)_{x \in X} \in Y^X$ .

We now characterize when the generalized Bernoulli shift is ergodic.

(2) $\Longrightarrow$ (1) It is straightforward.

(1) $\Longrightarrow$ (2) By assumption, the $\Gamma$ -action on $Y^X$ is ergodic and so all the orbits of the $\Gamma$ -action on X are infinite by (3). We have to show that the diagonal action of $\Gamma$ on $\prod_{i = 1}^n Y^X$ is ergodic for every integer $n \geq 1$ . For each n, this action is a generalized Bernoulli shift, where $\Gamma \curvearrowright Y^{\sqcup_{i = 1}^n X}$ simply by acting diagonally on the disjoint union of the sets X. Moreover, all the orbits of the $\Gamma$ -action on $\sqcup_{i = 1}^n X$ are infinite. Hence, now applying (3) $\Rightarrow$ (1) to the generalized Bernoulli shift $\Gamma \curvearrowright Y^{\sqcup_{i = 1}^n X}$ , we conclude that the action is ergodic, whence the thesis.

The following is an important criterion for the vanishing of bounded cohomology, which will be of key use for us.

2.3 Commuting conjugates

In the sequel, we are going to introduce sufficient conditions for $\mathfrak{X}$ -bounded acyclicity. Here, we recall the notion of commuting conjugates [Reference KotschickKot08, Reference Fournier-Facio and LodhaFFL23] that we will extend later.

Definition 2.19 (Commuting conjugates). A group $\Gamma$ has commuting conjugates if, for every finitely generated subgroup $H \leq \Gamma$ , there exists an element $g \in \Gamma$ such that

\begin{align*} [H, gHg^{-1}] = 1. \end{align*}

For simplicity, from now on we will always denote the conjugation by ${}^{g}H := gHg^{-1}$ .

This natural property detects bounded acyclicity in low degree.

1. (1) Let R be a ring. The infinite general linear group GL(R) has commuting conjugates [Reference Fournier-Facio, Löh and MoraschiniFFLM23, proof of Proposition 4.9].

2. (2) The group of symplectic (or Hamiltonian) diffeomorphisms of ${\mathbb{R}}^n$ with compact support and $n \geq 2$ has commuting conjugates. This result implicitly dates back to Kotschick’s work on the stable commutator length of the identity components of symplectic (or Hamiltonian) diffeomorphism groups [Reference KotschickKot08, § 4].

3. (3) Every group $\Gamma$ of boundedly supported homeomorphisms of the real line with no global fixpoints has commuting conjugates [Reference Fournier-Facio and LodhaFFL23, Proposition 3.8].

2.4 Direct unions

Here, we investigate the most important combination result for the proof of Theorem 1.3. To the best of the authors’ knowledge, it is not known whether direct unions of $\mathfrak{X}$ -boundedly acyclic groups are $\mathfrak{X}$ -boundedly acyclic (besides the case $\mathfrak{X}^{\text{dual}}$ , of course, corresponding to amenable groups). However, direct unions with controlled vanishing do preserve bounded acyclicity. This involves the notion of vanishing modulus with coefficients, which extends the previous notion for trivial real coefficients [Reference Fournier-Facio, Löh and MoraschiniFFLM23, Reference Monod and NarimanMN23]. The notion of vanishing modulus for trivial real coefficients has already proved useful to show that bounded acyclicity is preserved under certain constructions: namely if the vanishing modulus of a family of groups is uniformly bounded, then several constructions in bounded cohomology go through [Reference Fournier-Facio, Löh and MoraschiniFFLM23, LLM22]. Our aim is to extend this philosophy and make it quantitative in the case of coefficients. Let us start with the definition.

We apply the usual convention that $\inf \varnothing = \infty$ , so if $H^n_b(\Gamma; E) \neq 0$ , then $\| H^n_b(\Gamma; E) \| = \infty$ . An easy application of the open mapping theorem implies the converse.

1. (1) If $H^n_b(\Gamma; E) = 0$ , then $\| H^n_b(\Gamma; E) \| < +\infty$ .

2. (2) If $H^n_b(\Gamma; E) = 0$ for every $E \in \mathfrak{X}^{\text{sep}}$ , then $\| H^n_b(\Gamma; E) \|$ is uniformly bounded over all such E, i.e.

   \begin{align*} \sup_{E \in \mathfrak{X}^{\text{sep}}} \| H_b^n(\Gamma; E) \| < +\infty . \end{align*}

The first statement in the case of trivial real coefficients was already implicit in the work of Matsumoto and Morita [Reference Matsumoto and MoritaMM85] but the details have been worked out only recently [Reference Fournier-Facio, Löh and MoraschiniFFLM23, Lemma 4.12] (also compare with the work by Monod and Nariman [Reference Monod and NarimanMN23, after Definition 2.5]).

Proof. Ad (1) Let E be a Banach $\Gamma$ -module and let us suppose that $H^n_b(\Gamma; E)=0$ . Then, we can identify the space of n-cocycles with the space of n-coboundaries

\begin{align*}Z^n_b(\Gamma; E) = B^n_b(\Gamma; E) = \delta^{n-1}(C^{n-1}_b(\Gamma; E)^{\Gamma}). \end{align*}

Since this is a closed subspace of the Banach space $C^{n}_b(\Gamma; E)^{\Gamma}$ , it is itself a Banach space. Moreover, since $Z^{n-1}_b(\Gamma; E)$ is a closed subspace of the Banach space $C^{n-1}_b(\Gamma; E)^{\Gamma}$ there exists a natural quotient Banach space $C^{n-1}_b(\Gamma; E)^{\Gamma}/Z^{n-1}_b(\Gamma; E)$ endowed with the quotient norm. It follows that the bounded linear operator $\delta^{n-1}$ is a surjective continuous linear map between Banach spaces, so the open mapping theorem says that this is an open map. So the map

\begin{align*} \delta^{n-1} \colon C^{n-1}_b(\Gamma; E)^{\Gamma} / Z^{n-1}_b(\Gamma; E) \to Z^n_b(\Gamma; E) \end{align*}

has a bounded inverse $\varphi$ , which has finite operator norm $\| \varphi \|_{op}$ . Then, for every cocycle $z \in Z^{n}_b(\Gamma; E)$ , the class $\varphi(z)$ admits a representative $b \in C^{n-1}_b(\Gamma; E)^{\Gamma}$ of $\ell^\infty$ -norm at most $2 \cdot \|\varphi\|_{op} \cdot \| z \|_\infty$ . This shows that $\| H^n_b(\Gamma; E) \| < +\infty$ , whence the thesis.

Ad (2) Now suppose that $H^n_b(\Gamma; E) = 0$ for every Banach $\Gamma$ -module $E \in \mathfrak{X}^{\text{sep}}$ . Assume by way of contradiction that

\begin{align*}\sup_{E \in \mathfrak{X}^{\text{sep}}} \| H_b^n(\Gamma; E) \| =+\infty.\end{align*}

Then there exists a countable sequence of Banach $\Gamma$ -modules $E_i \in \mathfrak{X}^{\text{sep}}$ such that $\| H_b^n(\Gamma; E_i) \| \to +\infty$ when $i \to +\infty$ . Then, we can consider their $\ell^2$ -direct sum $E = \bigoplus_{2} E_i$ (see [Reference ConwayCon90, p. 72]). It is easy to show that E is a separable dual Banach $\Gamma$ -module [Reference ConwayCon90, Chapter III, Proposition 4.4, Exercises 4.13 and 5.4]. By our hypothesis $H^n_b(\Gamma; E) = 0$ , and thus $K := \| H^n_b(\Gamma; E) \| < +\infty$ by (1).

To achieve a contradiction, it suffices to show that $\| H^n_b(\Gamma; E_i) \| \leq K$ for all $i \in {\mathbb{N}}$ . Let us fix $i \in {\mathbb{N}}$ . We consider the inclusion $\iota \colon E_i \to E$ and the projection $\pi \colon E \to E_i$ . They are both $\Gamma$ -equivariant bounded maps [Reference ConwayCon90, Chapter III, Proposition 4.4]. Hence, the inclusion and the projection maps induce the following maps at the level of bounded cochain complexes:

\begin{align*}\pi_\ast^\bullet \colon C_b^\bullet(\Gamma; E)^\Gamma \to C_b^\bullet(\Gamma; E_i)^\Gamma \end{align*}

and

\begin{align*}\iota_\ast^\bullet \colon C_b^\bullet(\Gamma; E_i)^\Gamma \to C_b^\bullet(\Gamma; E)^\Gamma.\end{align*}

Let $z \in C^n_b(\Gamma; E_i)$ be a cocycle. We are going to show that we can find a primitive of norm at most $K \cdot \|z\|_\infty$ . First note that $\iota_*^n(z)\in C^n_b(\Gamma; E)^\Gamma$ is also a cocycle. By the definition of the vanishing modulus with coefficients in E and the definition of K, we know that there exists a cochain $b \in C^{n-1}_b(\Gamma; E)^{\Gamma}$ such that $\delta^{n-1}(b) = \iota_*^n(z)$ and $\| b \|_\infty \leq K \cdot \| \iota_*^n(z) \|_\infty$ . We then define $\pi_*^{n-1}(b) \in C^{n-1}_b(\Gamma; E_i)^{\Gamma}$ . By construction, we have that

\begin{align*}\| \pi_*^{n-1} (b) \|_\infty \leq \| b \|_\infty \leq K \cdot \| \iota_*^n (z) \|_\infty \leq K \cdot \| z \|_\infty.\end{align*}

Since $\delta^{n-1}(\pi^{n-1}_*(b)) = z$ , this proves our claim.

The reason why the vanishing modulus is relevant for us is the following result, which extends the corresponding statement for trivial real coefficients [Reference Fournier-Facio, Löh and MoraschiniFFLM23, Proposition 4.13] (in fact, that statement is only about direct unions, but for our purposes we need to treat slightly more general unions).

Proposition 2.24 (Vanishing modulus with coefficients and direct unions). Let I be an arbitrary index set and let $n \geq 1$ be an integer. Let $\Gamma$ be a group, and let $\{\Gamma_i\}_{i \in I}$ be a family of subgroups of $\Gamma$ such that each finitely generated subgroup of $\Gamma$ is contained in some $\Gamma_i$ . Let E be a Banach $\Gamma$ -module in $\mathfrak{X}^{\text{dual}}$ , and thus by restriction also a Banach $\Gamma_i$ -module for all $i \in I$ . Then

\begin{align*} \| H^n_b(\Gamma; E) \| \leq \sup_{i \in I} \| H^n_b(\Gamma_i; E) \|.\end{align*}

In particular, if the right-hand side is finite, then $H^n_b(\Gamma; E) = 0$ .

Proof. Let $K := \text{sup}_{i \in I} \| H^n_b(\Gamma_i; E) \|$ and suppose that $K < +\infty$ ; otherwise, there is nothing to show. Since $E \in \mathfrak{X}^{\text{dual}}$ , there exists a predual F with the corresponding $\Gamma$ -action. This also shows that $C^n_b(\Gamma; E) = \ell^\infty(\Gamma^{n+1}, E) \in \mathfrak{X}^{\text{dual}}$ , because it is the dual of $\ell^1(\Gamma^{n+1}, F)$ with the contragradient $\Gamma$ -action. In particular, it comes endowed with the weak- $\ast$ topology. Fix a cocycle $z \in Z^n_b(\Gamma; E)$ , and for each finitely generated subgroup $H \leq \Gamma$ define

\begin{align*} B_H^z := \{ b \in C^{n-1}_b(\Gamma; E)^{H} \mid \delta^{n-1}(b|_{H}) = z|_{H} \text{ and } \| b \|_\infty \leq K \cdot \| z \|_\infty \}. \end{align*}

It suffices to show that $\bigcap_{H \leq \Gamma \text{ fin. gen.}} B_H^z \neq \varnothing$ . Then, since $\Gamma$ is the direct union of its finitely generated subgroups, an element b in the intersection will satisfy:

• – $b \in C^{n-1}_b(\Gamma; E)^{\Gamma}$ ;

• – $\delta^{n-1}(b) = z$ ; and

• – $\| b \|_\infty \leq K \cdot \| z \|_\infty$ .

Now, each $B_H^z$ is a bounded weak- $\ast$ closed subset of $B_{K \cdot \|z\|_\infty}\subset C^{n-1}_b(\Gamma; E)$ . Hence, by the Banach–Alaoglu theorem, it suffices to show that the family $\{B_H^z\}_{H \leq \Gamma \text{ fin. gen.}}$ satisfies the finite intersection property.

Since $B_H^z \subset B_{H'}^z$ whenever $H' \leq H$ , and the system is directed, the finite intersection property will follow as soon as each $B_H^z$ is non-empty. Now, fix $H \leq \Gamma$ and let i be such that $H \leq \Gamma_i$ . Since $\| H^n_b(\Gamma_i; E) \| \leq K$ , there exists $b \in C^{n-1}_b(\Gamma_i, E)^{\Gamma_i}$ such that $\delta^{n-1}(b) = z|_{\Gamma_i}$ and $\| b \|_\infty \leq K \cdot \| z \|_\infty$ ; extending b by 0, we obtain an element in $B_H^z$ , which concludes the proof.

The nextroposition describes a way to obtain a uniform upper bound on a family of groups without actually having to estimate the individual vanishing moduli. It exploits Lemma 2.23.

Proposition 2.25 (Vanishing modulus with coefficients and direct sums). Let I be an arbitrary index set and let $n \geq 1$ be an integer. Let $\{\Gamma_i\}_{i \in I}$ be a family of groups and let $\Gamma := \bigoplus_{i \in I} \Gamma_i$ . Suppose that the classes of Banach $\Gamma_i$ -modules $\mathfrak{X}_i$ are in $\mathfrak{X}^{\text{sep}}$ for all $i \in I$ . Then, if $\Gamma$ is $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic, we have that

\begin{align*}\sup_{i \in I, E_i \in \mathfrak{X}_i} \| H^n_b(\Gamma_i; E_i) \| <+\infty.\end{align*}

A similar result for trivial real coefficients and unrestricted products has been stated by Monod and Nariman [Reference Monod and NarimanMN23, Proposition 2.6]. Here, we provide an elementary, self-contained proof, which allows for non-trivial coefficients as well. The core of the proposition lies in the following more general fact. Recall that a surjective homomorphism $r \colon \Gamma \to \Lambda$ is a retraction if there exists an injective homomorphism $\iota \colon \Lambda \to \Gamma$ such that $r \circ \iota = \text{id}_{\Lambda}$ .

Proof. Let $K := \| H^n_b(\Gamma; r^\ast E) \|$ and suppose that $K < +\infty$ ; otherwise, there is nothing to show. Let $z \in C^n_b(\Lambda; E)^\Lambda$ be a cocycle and let $C_b^n(r)(z) \in Z^n_b(\Gamma; r^\ast E)$ be its pullback along r. Then, by definition of K, there exists a cochain $b \in C_b^{n-1}(\Gamma; r^\ast E)^{\Gamma}$ such that $\delta^{n-1}(b) = C^n_b(r)(z)$ and $\| b \|_\infty \leq K \cdot \| C^n_b(r)(z) \|_\infty$ . Then the cochain $C^{n-1}_b(\iota) (b) \in C_b^{n-1}(\Lambda; E)^{\Lambda}$ satisfies

\begin{align*}\delta^{n-1} \circ C^{n-1}_b(\iota)(b) = C_b^n(\iota) \circ \delta^{n-1}(b) = C_b^n(\iota) \circ C^n_b(r) (z) = z \end{align*}

and

\begin{align*} \| C_b^{n-1}(\iota)(b) \|_\infty \leq \| b \|_\infty \leq K \cdot \| C_b^{n}(r)(z) \|_\infty \leq K \cdot \| z \|_\infty. \end{align*}

This shows that $\| H^n_b(\Lambda; E) \| \leq K$ , whence the thesis.

Proof of Proposition 2.25. By Lemma 2.23(2), we know that there exists a uniform constant K such that $\| H^n_b(\Gamma; E) \| \leq K$ for every Banach $\Gamma$ -module E in $\mathfrak{X}^{\text{sep}}$ . In particular, for all $i \in I$ and for all Banach $\Gamma_i$ -modules $E_i\in\mathfrak{X}_i$ , letting $\pi_i \colon \Gamma \to \Gamma_i$ denote the corresponding retraction (that is, the projection onto the ith factor), Lemma 2.26 implies that $\| H^n_b(\Gamma_i; E_i) \| \leq \| H^n_b(\Gamma; \pi_i^\ast E_i) \| \leq K$ . Since $i \in I$ and $E_i$ were arbitrary chosen, the thesis follows.

3.1 Permutational wreath products

We extend Monod’s construction for computing the bounded cohomology of wreath products to the case of permutational wreath products.

Let $\Gamma$ and A be groups and let X be an A-set. The permutational (restricted) wreath product $\Gamma\wr_X A$ is defined as $(\bigoplus_{x\in X}\Gamma)\rtimes A$ , where A acts on the subgroup $\bigoplus_{x\in X}\Gamma$ by permutation of the factors. The case $X = A$ with the regular action recovers the usual regular wreath product, or simply wreath product, denoted by $\Gamma\wr A$ .

Setup 3.1 (Permutational wreath products with amenable acting group). We consider the following setup.

1. (1) Let $\Gamma$ and A be two countable groups, with A amenable.

2. (2) Let X be a countable A-set such that all the A-orbits are infinite.

3. (3) Let $(Y_0, \mu_0)$ be the standard Borel probability $\Gamma$ -space obtained by endowing $\Gamma$ with a distribution of full support (so $Y_0 = \Gamma$ as a set).

4. (4) Let $Y := Y_0^X$ be the standard Borel probability space endowed with the product measure $\mu$ .

5. (5) The permutational wreath product $\Gamma \wr_X A$ acts on Y as follows: $\bigoplus_{x \in X} \Gamma$ acts coordinate-wise and A acts by shifting the coordinates.

An analogous space has been considered by Monod [Reference MonodMon22, § 2.3] in the case of regular wreath products, and the generalization is straightforward. We have the following.

We are left to show that the action is also Zimmer-amenable. By Proposition 2.10(1), the space $Y_0$ is a Zimmer-amenable $\Gamma$ -space. Moreover, Proposition 2.10(2) shows that Y is a Zimmer-amenable $\bigoplus_{x \in X} \Gamma$ -space (because X is countable). Since the base group $\bigoplus_{x \in X} \Gamma$ is co-amenable in $\Gamma \wr_X A$ (because A is amenable), by Proposition 2.10(3), we conclude that Y is a Zimmer-amenable $\Gamma \wr_X A$ -space.

3.2 Vanishing modulus with coefficients

Quite surprisingly, the vanishing result in Corollary 3.4 is very controlled. Our next goal is to make this precise. This fact will be at the heart of the proof of Theorem 1.3:

Let $\Gamma$ , X, and A be as in Setup 3.1. Let $\Lambda$ be a quotient of $\Gamma \wr_X A$ by a normal amenable subgroup N. We then have $\| H^n_b(\Lambda; E) \| \leq K_n$ for every separable dual Banach $\Lambda$ -module E.

Proof. Let $n \geq 1$ be an integer. Suppose by contradiction that such a $K_n < +\infty$ does not exist. We can then find a countable sequence

\begin{align*}\{\Lambda_i = \Gamma_i \wr_{X_i} A_i / N_i\}_{i \in{\mathbb{N}}}, \end{align*}

where each $\Gamma_i \wr_{X_i} A_i$ is as in Setup 3.1, and $N_i$ is a normal amenable subgroup; and for each $i \in {\mathbb{N}}$ a separable dual Banach $\Lambda_i$ -module $E_i$ such that

\begin{align*}\sup_{i \in {\mathbb{N}}} \| H^n_b(\Lambda_i; E_i) \| =+\infty. \end{align*}

Once the claim is proved, it will be in contradiction with Proposition 2.25. This will imply that the original assumption was false and the required $K_n<+\infty$ exists.

Proof of the claim. Since the kernel of the quotient map $\bigoplus_{i \in {\mathbb{N}}} (\Gamma_i \wr_{X_i} A_i) \to \bigoplus_{i \in {\mathbb{N}}} \Lambda_i$ is the amenable group $\bigoplus_{i \in {\mathbb{N}}} N_i$ (being the direct sum of amenable groups), using Theorem 2.6 it suffices to show the $\mathfrak{X}^{\text{sep}}$ -bounded acyclicity of the group $G := \bigoplus_{i \in {\mathbb{N}}} (\Gamma_i \wr_{X_i} A_i)$ .

We are going to show that there exists a $\text{HEZA}$ -space for G. Let Y be G itself endowed with a distribution of full support. This is a Zimmer-amenable space for G, and for each subgroup $\Gamma_i$ , by Proposition 2.10(1). Hence for each $i \in {\mathbb{N}}$ , we can construct a generalized Bernoulli shift $A_i \curvearrowright Y^{X_i}$ and obtain an action of $\Gamma_i \wr_{X_i} A_i$ on $Y^{X_i}$ as explained in Setup 3.1. Proposition 3.2 still applies in this case (we have only changed our space Y in Setup 3.1) and so we have that for each $i \in {\mathbb{N}}$ , the standard Borel probability space $Y^{X_i}$ is a $\text{HEZA}$ -space for $\Gamma_i \wr_{X_i} A_i$ .

We now consider the standard Borel probability space $\prod_{i \in {\mathbb{N}}} Y^{X_i}$ endowed with the product measure. We let G act on it coordinate-wise. By Proposition 2.10(2), this action is still non-singular and Zimmer-amenable. We are left to prove that the action $G \curvearrowright \prod_{i \in {\mathbb{N}}} Y^{X_i}$ is also highly ergodic.

Let us consider the subgroup $\bigoplus_{i \in {\mathbb{N}}} A_i \leq G$ . By construction, this group acts on $\prod_{i \in {\mathbb{N}}} Y^{X_i} = Y^{\sqcup_{i \in {\mathbb{N}}} X_i}$ by the shifting coordinate-wise action $\bigoplus_{i \in {\mathbb{N}}} A_i \curvearrowright \sqcup_{i \in {\mathbb{N}}}X_i$ . Hence by the hypothesis of Setup 3.1, all the orbits are infinite (since the orbits of each action $A_i \curvearrowright X_i$ are so), the group $\bigoplus_{i \in {\mathbb{N}}} A_i$ is countable, and the set $\sqcup_{i\in {\mathbb{N}}} X_i$ is as well. According to Proposition 2.15, this new generalized Bernoulli shift is highly ergodic. Since $\bigoplus_{i \in {\mathbb{N}}}A_i$ is a subgroup of G, Remark 2.16 shows that the action of G on $\prod_{i \in {\mathbb{N}}} Y^{X_i}$ is also highly ergodic. By Corollary 2.18, G is $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic, whence the desired contradiction.

4.1 Commuting cyclic conjugates and $\mathfrak{X}^{\text{sep}}$ -bounded acyclicity

We recall the definition of commuting cyclic conjugates from § 1.

It will also be useful to treat the special situation in which $n = \infty$ .

Our main result is the following more precise version of Theorem 1.3.

1. (1) If $\Gamma$ is a group with commuting cyclic conjugates, then it is $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic.

2. (2) For every integer $n \geq 1$ , there exists a constant $K_n \in [0, \infty)$ such that for every group $\Gamma$ with commuting cyclic conjugates, and every separable dual Banach $\Gamma$ -module E, we have that $\| H^n_b(\Gamma; E) \| \leq K_n$ . Hence, the vanishing modulus is uniform.

Our result is built on Proposition 3.5 and so we have to rephrase the definition of commuting cyclic conjugates in terms of morphisms from wreath products. This is the content of the following technical proposition.

1. (1) The group $\Gamma$ has commuting cyclic conjugates.

2. (2) For every finitely generated subgroup $H \leq \Gamma$ , there exist a countable amenable group A, an infinite-index subgroup $B < A$ , and a homomorphism $f \colon A \to \Gamma$ such that: 8pt

   1. (i) $[H, {}^{f(a)}H] = 1$ for all $a \in A \setminus B$ ; and

   2. (ii) $[H, f(b)] = 1$ for all $b \in B$ .

3. (3) For every finitely generated subgroup $H \leq \Gamma$ , there exist a countable amenable group A, an infinite transitive A-set X, and a homomorphism

   $$f\colon H \wr_X A \to \Gamma$$
   such that f restricts to the inclusion on some factor H, and f has amenable kernel.

We postpone the proof of Proposition 4.5 to § 4.2 and here we give the proof of Theorem 4.4 (and thus of Theorem 1.3).

Proof of Theorem 4.4. Let $\Gamma$ be a group with commuting cyclic conjugates, and let $n \geq 1$ . Let $H \leq \Gamma$ be a finitely generated subgroup. By Proposition 4.5, there exist a countable amenable group A, an infinite transitive A-set X (which is necessarily countable), and a morphism $f \colon H \wr_X A \to \Gamma$ that restricts to the inclusion on some factor H. In particular, we have that $H \leq \textrm{im}(f)$ . Again by Proposition 4.5, we can assume that the kernel of f is amenable. Let $\mathfrak{X}^{\text{sep}}$ be the class of separable dual Banach $\Gamma$ -modules and let $i^*(\mathfrak{X}^{\text{sep}})$ be the class of separable dual Banach $\textrm{im}(f)$ -modules induced by the inclusion $i \colon \textrm{im}(f) \to \Gamma$ .

We are now ready to apply Proposition 3.5. Indeed, H is finitely generated and thus countable, X is countably infinite, and the amenable group A acts on X with a (unique) infinite orbit. Hence, for every integer $n \geq 1$ , there exists a constant $K_n \in [0, \infty)$ that does not depend on H, A, X, or even $\Gamma$ , such that $\| H^n_b(\textrm{im}(f); E) \| \leq K_n$ for every separable dual Banach $\textrm{im}(f)$ -module E.

Every finitely generated $H \leq \Gamma$ is contained in a group of the form $\textrm{im}(f)$ as above. Hence, applying Proposition 2.24 with $\Gamma_i = \textrm{im}(f)$ , we conclude that $H^n_b(\Gamma; E) = 0$ .

4.2 Proof of Proposition 4.5

We prove the chain of implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (1). The first implication is the crucial one.

Proof of Proposition 4.5 (1) $\Rightarrow$ (2). Suppose that $\Gamma$ has commuting cyclic conjugates, and let $H \leq \Gamma$ be finitely generated. Set $\Lambda_0 := H$ . Let $t_1 \in \Gamma$ and $n_1 \in {\mathbb{Z}}_{\geq 2} \cup \{ \infty \}$ be such that:

• – $[H, {}^{t_1^p}H]=1$ for all $1 \leq p < n_1$ ; and

• – $[H, t_1^{n_1}] = 1$ .

Define $\Lambda_1 := \langle H, t_1 \rangle$ . We proceed by induction. At the ith step, $i \in {\mathbb{N}}$ , we have a finitely generated group $\Lambda_{i-1}$ , and we find $t_{i} \in \Gamma$ and $n_{i} \in {\mathbb{Z}}_{\geq 2} \cup \{ \infty \}$ such that:

• – $[\Lambda_{i-1}, {}^{t_i^p}\Lambda_{i-1}]=1$ for all $1 \leq p <n_i$ ; and

• – $[\Lambda_{i-1}, t_{i}^{n_{i}}] = 1$ .

We set $\Lambda_{i} := \langle \Lambda_{i-1}, t_{i} \rangle$ .

First, suppose that $n_i = \infty$ for some $i \in {\mathbb{N}}$ . In that case, for all integers $p \geq 1$ , we have

\begin{align*}[H, {}^{t_i^p}H] \leq [\Lambda_{i-1}, {}^{t_i^p}\Lambda_{i-1}] = 1, \end{align*}

and, moreover, for all integers $p \leq -1$ , we have

\begin{align*}{}^{t_i^{-p}}[H, {}^{t_i^p}H] = [{}^{t_i^{-p}} H, H] = [H, {}^{t_i^{-p}}H]^{-1} = 1 \Longrightarrow [H, {}^{t_i^p}H] = 1. \end{align*}

Thus we can choose $A = {\mathbb{Z}}, B = \{ 0 \}$ and a homomorphism

\begin{align*} f \colon A = {\mathbb{Z}} &\to \Gamma \\ m &\mapsto t_i^m,\end{align*}

which satisfies the conditions required in item (2). Note how this already suffices to show that groups with commuting $ {\mathbb{Z}}$ -conjugates satisfy condition (2). From now on, we may assume that $n_i \in {\mathbb{Z}}_{\geq 2}$ for all integers $i \geq 1$ .

Set $A_1 := {\mathbb{Z}}$ , and define by induction

\begin{align*} A_{i+1} := A_i \wr_{{\mathbb{Z}}/n_{i+1}} {\mathbb{Z}}, \end{align*}

where the action of ${\mathbb{Z}}$ on ${\mathbb{Z}}/n_{i+1}$ is by left translation. By construction, each $A_i$ is amenable, being an extension of amenable groups. Consider the inclusion of $A_i$ into $A_{i+1}$ , concentrated at the coordinate $0 \in {\mathbb{Z}}/n_{i+1}$ . Let A be the direct union of the $A_i$ . Hence, the group A is countable and amenable, since it is a direct union of countable amenable groups.

Let $B_1 := n_1 {\mathbb{Z}} \leq A_1$ , and define by induction $B_{i+1}$ to be the subgroup of $A_{i+1}$ consisting of elements of the form $((a_p)_{p \in {\mathbb{Z}}/n_{i+1}}, m) \in A_i \wr_{{\mathbb{Z}}/n_{i+1}} {\mathbb{Z}}$ , where $a_0 \in B_i$ and $m \in n_{i+1} {\mathbb{Z}}$ . This is indeed a group, since the ${\mathbb{Z}}$ coordinate only consists of multiples of $n_{i+1}$ , which act trivially on ${\mathbb{Z}}/n_{i+1}$ . Denote by B the direct union of the $B_i$ . By construction, $B < A$ .

To see that the subgroup $B < A$ has infinite index, consider the A-conjugates of $A_1$ . First note that there are exactly $n_i$ conjugates of $A_{i-1}$ inside $A_i$ , which have pairwise trivial intersection: these are the copies of $A_{i-1}$ concentrated at the different coordinates of ${\mathbb{Z}}/n_i$ . It follows that the number of $A_i$ -conjugates of $A_1$ is at least $n_2 \cdots n_i$ . Since, by definition of commuting cyclic conjugates, each $n_i \geq 2$ (Definition 4.1), there are infinitely many A-conjugates of $A_1$ . On the other hand, let us show that B centralizes $A_1$ . This is clear for $B_1$ . Recall that an element of $B_{i+1}$ is of the form $((a_p)_{p \in {\mathbb{Z}}/n_{i+1}}, m)$ , where $a_0 \in B_i$ and $m \in n_{i+1}{\mathbb{Z}}$ . Then $a_0$ centralizes $A_1$ by the induction hypothesis, and $a_p$ with $p \neq 0$ as well as m centralize $A_1$ by definition of the permutational wreath product. Thus all of $B_{i+1}$ centralizes $A_1$ and so their direct union B also centralizes $A_1$ . In conclusion, we have proved that the conjugation action of A on the set of its subgroups has the following properties:

1. (1) the A-conjugates of $A_1$ are infinite (i.e. the orbit ${}^{A}A_1$ is infinite); and

2. (2) B centralizes $A_1$ (i.e. B is a subgroup of the stabilizer $\text{Stab}_A(A_1)$ ).

The orbit-stabilizer theorem then shows that

\begin{align*}\left[ A : B\right] \geq \left[ A : \text{Stab}_A(A_1)\right] = |{}^{A} A_1| = +\infty, \end{align*}

and thus B has infinite index in A.

We are left to define the desired homomorphism $f \colon A \to \Gamma$ as required in condition (2). We also construct the homomorphism f by induction. Let us begin by setting

\begin{align*} f \colon A_1 = {\mathbb{Z}} &\to \Gamma \\ m &\mapsto t_1^m.\end{align*}

By definition, we have $f(A_1) \leq \Lambda_1$ . Now suppose that f has already been defined on $A_i$ (and is such that $f(A_i) \leq \Lambda_{i}$ ), and set

\begin{align*} f \colon A_{i+1} = A_i \wr_{{\mathbb{Z}}/n_{i+1}} {\mathbb{Z}} &\to \Gamma \\ ((a_p)_{p \in {\mathbb{Z}}/n_{i+1}}, m) &\mapsto \bigg(\prod_{p = 0}^{n_{i+1}-1} {}^{t_{i+1}^p}f(a_p) \bigg)t_{i+1}^m.\end{align*}

Let us check that this is a homomorphism. By induction, it is a homomorphism on each summand $A_i$ . Recall that $\langle t_{i+1}^{n_{i+1}} \rangle$ centralizes $\Lambda_i$ and therefore $f(A_i)$ (since $f(A_i) \leq \Lambda_i$ by induction). Also recall that $[f(A_i), {}^{t_{i+1}^p}f(A_i)] \leq [\Lambda_i, {}^{t_{i+1}^p}\Lambda_i] = 1$ for all $1 \leq p < n_{i+1}$ . Applying a suitable conjugation, we see that the subgroups ${}^{t_{i+1}^p}f(A_i)$ pairwise commute; hence f is a homomorphism on the direct sum of the $A_i$ . Finally, the action of ${\mathbb{Z}}$ by conjugacy on the summands is respected, by construction and because $t_{i+1}^{n_{i+1}}$ centralizes each summand.

We now need to check the two required properties in condition (2):

1. (i) $[H, {}^{f(a)} H] = 1$ for all $a \in A \setminus B$ ; and

2. (ii) $[H, f(b)] = 1$ for all $b \in B$ .

Property (ii) is verified by induction on the $B_i$ , with the same argument as the proof above that B centralizes $A_1$ together with the definition of f.

Let us prove that property (i) is also satisfied. We again proceed by induction, by showing that if $a \in A_i$ is such that $[H,{}^{f(a)} H] \neq 1$ , then $a \in B_i$ . For $i = 1$ , this is by definition of $t_1$ . Now suppose that $a \in A_{i+1}$ is an element such that $[H, {}^{f(a)} H] \neq 1$ . Using the definition of $A_{i+1}= A_i \wr_{{\mathbb{Z}}/n_{i+1}} {\mathbb{Z}}$ , we write $a =((a_p)_{p \in {\mathbb{Z}}/n_{i+1}}, m)$ . Now, for $1 \leq p < n_{i+1}$ , the image of $a_p$ lies in ${}^{t_{i+1}^p}f(A_i) \leq {}^{t_{i+1}^p}\Lambda_i$ , which by choice of $t_{i+1}$ commutes with $\Lambda_i$ , and therefore with H. In other words, conjugating by these coordinates has no effect on H, and therefore ${}^{f(a)^{-1}} H = {}^{t_{i+1}^{-m}}({}^{f(a_0)^{-1}}H)$ . First, suppose that $m \notin n_{i+1} {\mathbb{Z}}$ . Then ${}^{t_{i+1}^{-m}}({}^{f(a_0)^{-1}}H) \leq {}^{t_{i+1}^{-m}}\Lambda_i$ , which commutes with $\Lambda_i$ and thus with H. A standard manipulation then shows that ${}^{f(a)} H$ also commutes with H. Since we are assuming that $[H, {}^{f(a)} H] \neq 1$ , we obtain a contradiction. Therefore, we must have $m \in n_{i+1} {\mathbb{Z}}$ , which implies that ${}^{t_{i+1}^{-m}}({}^{f(a_0)^{-1}}H) = {}^{f(a_0)^{-1}}H$ , since $t_{i+1}^{n_{i+1}}$ centralizes $\Lambda_i$ . Since we are assuming $[H, {}^{f(a)}H] \neq 1$ , or equivalently $[H, {}^{f(a)^{-1}}H] \neq 1$ , we obtain $[H, {}^{f(a_0)^{-1}}H] \neq 1$ and so by the induction hypothesis, $a_0$ actually lies in $B_i$ . We have shown that $a = ((a_p)_{p \in {\mathbb{Z}}/n_{i+1}}, m)$ is such that $a_0 \in B_i$ and $m \in n_{i+1}{\mathbb{Z}}$ . This shows that $a \in B_{i+1}$ by definition, and concludes the proof.

Proof of Proposition 4.5 (2) $\Rightarrow$ (3). Let $H \leq \Gamma$ be a finitely generated subgroup, and suppose that there exists a countable amenable group A, a morphism $f \colon A \to \Gamma$ , and an infinite index subgroup $B < A$ as in condition (2). Note that the property (2)(i) implies that $[^{f(a_1)} H, ^{f(a_2)} H] = 1$ whenever $a_1B \neq a_2B$ . Define $X := A/B$ , with the left-translation action of A. By the assumptions of (2), X is a transitive infinite A-set. We extend f to a homomorphism $f \colon H \wr_X A \to \Gamma$ as follows:

\begin{align*} f((h_{aB})_{aB \in X}, g) = \bigg(\prod_{aB \in X} {}^{f(a)}h_{aB}\bigg)f(g). \end{align*}

Since f(B) centralizes H, the value of $^{f(a)}h_{aB}$ is independent of the choice of a. Moreover, since all of the $^{f(a)}H$ pairwise commute, when a runs over a system of representatives, the product is well defined independently of the order of multiplication. Finally, the action by conjugacy of A on the H summands corresponds equivariantly to the action by conjugacy of f(A) on the commuting groups ${}^{f(a)}H : aB \in X$ . This implies that f is indeed a homomorphism, and by construction it restricts to the inclusion of H on the factor $H_{1B}$ . This proves the first part of the claim.

We are left to show that f has amenable kernel. A proof in the case $A = X = {\mathbb{Z}}$ is already available in the literature [Reference Fournier-Facio and RangarajanFFR24, Lemma 5.4]. For completeness, we show how to extend it. Let $f \colon H \wr_X A \to \Gamma$ be as above, and let $x_0 \in X$ be such that $f|_{H_{x_0}}$ is the inclusion of H. Note that the structure of the wreath product implies that $\ker(f)$ can be described as the following extension:

\begin{align*} 1 \to K := \ker(f) \cap \bigoplus_{x \in X} H \to \ker(f) \to Q \to 1, \end{align*}

where Q is the subgroup of A corresponding to the projection of $\ker(f)$ to A. It is sufficient to show that K is amenable; in fact, we will show that it is abelian.

To see this, let $g, h \in K$ , which we write as $g = (g_x)_{x \in X}, h = (h_x)_{x \in X}$ (we omit the A-component since it is always trivial). Then,

\begin{align*} 1 = f(g) = \prod_{x \in X} f(g_x) \quad \Longrightarrow \quad f(g_{x_0})^{-1} = \prod_{x \in X \setminus \{ x_0 \}} f(g_x). \end{align*}

Since $f(H_x)$ commutes with $f(H_{x_0})$ for all $x \in X \setminus \{ x_0 \}$ , from the above expression it follows, in particular, that $f(g_{x_0})$ commutes with $f(h_{x_0})$ . But $f|_{H_{x_0}}$ is injective by hypothesis; therefore $g_{x_0}$ and $h_{x_0}$ commute. Up to conjugating by a suitable element of A, we obtain that $g_x$ commutes with $h_x$ for all $x \in X$ , and therefore that g and h commute.

Proof of Proposition 4.5 (3) $\Rightarrow$ (1). Suppose that $\Gamma$ is a group satisfying condition (3). Let $H \leq \Gamma$ be a finitely generated group and let $f\colon H \wr_X A \to \Gamma$ be as required by condition (3). Let $x_0 \in X$ be such that f restricts to the identity on the $x_0$ -coordinate. We set $t := f(a)$ for some $a \in A$ such that $a \cdot x_0 \neq x_0$ . Let $n \geq 2$ be the minimal integer such that $a^n \cdot x_0 = x_0$ (possibly $n = \infty$ ). Then, for all $1 \leq p < n$ , we have $[H, {}^{t^p}H] = [f(H_{x_0}), {}^{f(a^p)}f(H_{x_0})] = f([H_{x_0}, H_{a^p \cdot x_0}]) = 1$ , because $a^p \cdot x_0 \neq x_0$ ; and $[H, t^n] = f([H_{x_0}, a^n]) = 1$ , since $a^n \cdot x_0 = x_0$ . Hence, $\Gamma$ has commuting cyclic conjugates.

4.3 Constructions

A classical question in bounded cohomology with real coefficients is whether surjective group homomorphisms induce injective maps in bounded cohomology [Reference BouarichBou01, Reference Capovilla, Löh and MoraschiniCLM22, Reference Fournier-Facio, Löh and MoraschiniFFLM24, Reference Fournier-Facio, Löh and MoraschiniFFLM23]. Since in this paper we are working with $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic groups, a weaker formulation of the previous problem is whether quotients of $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic groups are $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic. For this reason, now that we have provided a new criterion for bounded acyclicity, it is interesting to note that it does pass to quotients. This shows that quotients of groups with commuting cyclic conjugates are $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic.

Proof. Let $\Gamma$ be a group, and let $\pi \colon \Gamma \to\overline{\Gamma}$ be a quotient. For a finitely generated subgroup $\overline{H} \leq \overline{\Gamma}$ , choose a preimage for each of the generators to obtain a finitely generated subgroup $H \leq \Gamma$ such that $\pi(H) = \overline{H}$ . Since $\Gamma$ has commuting cyclic conjugates, there exist $t \in \Gamma$ and $n \in {\mathbb{Z}}_{\geq 2} \cup \{ \infty \}$ such that $[H, {}^{t^p}H] = 1$ for $1 \leq p < n$ and $[H, t^n] = 1$ . Then, by definition of homomorphism, we have

\begin{align*}[\overline{H}, {}^{\pi(t)^p}\overline{H}] = \pi([H, {}^{t^p}H]) = 1 \end{align*}

for all $1 \leq p < n$ ; and

\begin{align*}[\overline{H}, \pi(t)^n] = \pi([H, t^n]) = 1. \end{align*}

Thus $\overline{H}$ satisfies the condition of commuting cyclic conjugates with $\pi(t)$ and n. Note that n is unchanged, so this also proves that if $\Gamma$ has commuting ${\mathbb{Z}}$ -conjugates, then $\overline{\Gamma}$ also has commuting ${\mathbb{Z}}$ -conjugates.

Extensions of $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic groups can be shown to be $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic, using either spectral sequences [Reference MonodMon01, § 12] or generalized mapping theorems [Reference Moraschini and RaptisMR23]. For groups with commuting $\mathbb{Z}$ -conjugates, we can show more.

For direct sums, there exists a finite index set $J \subset I$ such that $H_i$ is trivial for all $i \in I \setminus J$ . So we can choose $t_i = 1$ for all $i \in I \setminus J$ , and the resulting element t belongs to the direct sum.

We use this to prove an extension of the uniformity of the vanishing modulus from Theorem 4.4, which will be useful in the sequel.

We end by noting that the commuting cyclic conjugates condition passes to derived subgroups. Recall that for a group $\Gamma$ , we denote $\Gamma^{(1)} := [\Gamma, \Gamma]$ and $\Gamma^{(d)} := [\Gamma^{(d-1)}, \Gamma^{(d-1)}]$ for all $d \gt 1$ .

5.1 Stable mapping class group

First, we show that the stable mapping class group $\Gamma_\infty=\cup_{g\geq 1}\Gamma_g^1$ has commuting cyclic conjugates and so it is $\mathfrak{X}^{\text{sep}}$ -boundedly acyclic, proving a conjecture by Bowden [Reference BowdenBow11].

Note that $\Gamma_\infty$ is perfect [Reference Farb and MargalitFM12, Theorem 5.2].

Figure 1.

The boundary connected sum of $\Sigma_2^1$ and $S_2^1$ along the lower half of their boundaries.

We claim that there exists an element $t \in \Gamma_{2g}^1 \leq \Gamma_\infty$ such that:

1. (1) $[H, {}^t H] = 1$ ; and

2. (2) $[H, t^2] = 1$ .

Since every element h in $H \leq \Gamma_g^1$ fixes the boundary, the collar neighbourhood theorem ensures that one can find a small enough neighbourhood $S^1 \times [0, 1)$ of the boundary of $\Sigma_{2g}^1$ such that the support of every $h \in H$ is contained in $\Sigma_g^1 \cap (\Sigma_{2g}^1 \setminus (S^1 \times [0, 1)))$ . Now let $t \in \Gamma_{2g}^1$ be the half turn around the curve $S^1\times \{3/4\}$ over $\Sigma_{2g}^1 \setminus (S^1 \times [0, 1/2])$ and extend it to the rest of $\Sigma_{2g}^1$ by fixing a neighbourhood $S^1\times [0,1/4]$ of its boundary (Figure 2). This t thus acts as a central symmetry below the curve $S^1\times \{3/4\}$ and rigidly exchanges $\Sigma^1_g\cap (\Sigma_{2g}^1 \setminus (S^1 \times [0, 1)))$ and $S^1_g\cap (\Sigma_{2g}^1 \setminus (S^1 \times [0, 1)))$ . By construction, for every $h \in H$ , the conjugate element ${}^t h$ has its compact support entirely contained in the interior of $S_g^1$ and so it commutes with h. This shows that $[H, {}^tH] = 1$ .

Figure 2.

The element $t\in \Gamma_{2g}^1$ is a half turn along the middle curve in the direction of the arrows.

On the other hand, if we consider the Dehn twist $t^2$ , we obtain an element of $\Gamma_{2g}^1$ whose compact support is entirely contained in the open neighbourhood $S^1 \times (1/4, 3/4)$ . Since we have chosen a small enough collar neighbourhood, we have that $t^2$ has support disjoint from that of every $h \in H$ . Thus condition (2) is also satisfied and we have shown that $\Gamma_\infty$ has commuting cyclic conjugates. By applying Theorem 1.3, we obtain the result.

5.2 Other compactly supported big mapping class groups

We have studied the group $\Gamma_\infty$ as a natural direct limit of finite type mapping class groups. One could take another point of view, coming from the theory of big mapping class groups [Reference Aramayona and VlamisAV20]. For a connected orientable surface $\Sigma$ of infinite type, i.e. with infinitely generated fundamental group, we denote by $\text{MCG}(\Sigma)$ the group of orientation-preserving homeomorphisms of $\Sigma$ , modulo those that are isotopic to the identity. The compactly supported subgroup is the group $\text{MCG}_c(\Sigma)$ of mapping classes that admit a representative with compact support. With this language, the group $\Gamma_\infty$ is nothing but $\text{MCG}_c(\Sigma_{LN})$ , where $\Sigma_{LN}$ is the Loch Ness monster surface (see Figure 3).

The key ingredient in the proof of Corollary 5.1 is that every subsurface $K \subset \Sigma_{LN}$ of finite type can be displaced by a compactly supported mapping class (with additional control on powers of the displacing element). This is necessary for such a result to hold, as is shown by a criterion due to Horbez, Qing, and Rafi [HQR22]. For its statement, we need some terminology. A subsurface of finite type $K \subset \Sigma$ is non-displaceable if $\varphi(K) \cap K \neq \varnothing$ for all $\varphi \in \mathrm{Homeo}(\Sigma)$ . Let $\widehat{K}$ denote the surface obtained from K by capping off each boundary component with a once-punctured disc. A subgroup $G < \text{MCG}(\Sigma)$ is said to be K-non-elementary if the stabilizer of K in G contains two elements that induce independent pseudo-Anosov mapping classes of $\widehat{K}$ .

In particular, such a G does not have commuting cyclic conjugates, or even commuting conjugates. This applies to $G = \text{MCG}_c(\Sigma)$ , whenever $\Sigma$ contains a sufficiently complicated non-displaceable subsurface K.

As a special instance, we consider alternative stabilizations of mapping class groups, in the same spirit as the previous construction of $\Gamma_\infty$ . We attach k copies of the two-holed torus to a sphere with k boundary components, one on each of its boundary components. At each step, this produces an embedding $\Sigma_{kg, k} \hookrightarrow \Sigma_{k(g+1), k}$ , whence an inclusion of the mapping class groups $\Gamma_{kg}^k \hookrightarrow \Gamma_{k(g+1)}^k$ .

We denote by $\Gamma_{\infty, k}$ the direct union of these groups. We can see $\Gamma_{\infty, k}$ as $\text{MCG}_c(\Sigma_{\infty, k})$ , where $\Sigma_{\infty, k}$ is the limit of the surfaces $\Sigma_{kg, k}, g \to \infty$ .

When $k = 1$ , this is just $\Gamma_\infty$ , and $\Sigma_{\infty, 1} = \Sigma_{LN}$ is the Loch Ness monster surface. When $k \geq 3$ , Theorem 5.2 applies, and shows that $\Gamma_{\infty, k}$ has infinite-dimensional second bounded cohomology. Indeed, the subsurfaces $\Sigma_{kg, k}$ are all non-displaceable, since $\Sigma_{\infty, k} \setminus \Sigma_{kg, k}$ has $k \geq 3$ connected components, while the complement of any subsurface disjoint from $\Sigma_{kg, k}$ has at most two connected components.

An interesting intermediate case is given by $k = 2$ , when $\Sigma_{\infty, 2} = \Sigma_{JL}$ is also known as the Jacob’s ladder surface (see Figure 3). Then, every subsurface of finite type is displaceable by a mapping class, which can be chosen to be a vertical translation; so Theorem 5.2 does not apply. However, it is not possible to achieve this displacement using a compactly supported mapping class, i.e. an element of $\Gamma_{\infty, 2} = \text{MCG}_c(\Sigma_{JL})$ . Indeed, let $K \cong \Sigma_{2g,2}$ be a subsurface of $\Sigma_{JL}$ . Then, an element $t \in \Gamma_{\infty, 2}$ must be represented by a homeomorphism supported on a subsurface $K' \cong \Sigma_{2h, 2}$ containing K, for some $h \geq g$ . The complement $K' \setminus K$ is the disjoint union of two subsurfaces $\Sigma_{2g_1,2} \sqcup \Sigma_{2g_2, 2}$ , and $t(K' \setminus K) = K' \setminus t(K)$ must be of the same type. It is easy to see that it is not possible to preserve the genera $g_1, g_2$ , while ensuring that $t(K) \subset K'$ is disjoint from K. Contrast this with the case of the Loch Ness monster, which is illustrated in Figure 2.

Figure 3.

Two infinite-type surfaces: (a) the Loch Ness monster and (b) Jacob’s ladder.

Camille Horbez has informed us that it should be possible to generalize Theorem 5.2 to include surfaces K that are only G-non-displaceable, i.e. such that $\varphi(K) \cap K \neq \varnothing$ for every homeomorphism $\varphi \in \text{Homeo}_c(\Sigma)$ representing an element of G. The above argument would then imply that $\Gamma_{\infty, 2}$ has infinite-dimensional second bounded cohomology.

5.3 Stable braid group and the bounded cohomology of the braided Ptolemy–Thompson group

In this section, we compute the bounded cohomology of the infinite braid group $B_\infty$ (with compact support). Kotschick has proved that $B_\infty$ has vanishing stable commutator length [Reference KotschickKot08]. We adapt his argument to show that it has, in fact, commuting cyclic conjugates. As a by-product, we fully compute the bounded cohomology of the braided Ptolemy–Thompson group $T^\ast$ (see [Reference Funar and KapoudjianFK08]).

The commutator subgroup $[B_\infty, B_\infty]$ has been studied on some occasions [Reference Burago, Ivanov and PolterovichBIP08, BK15, Reference KimuraKim18]; note that it is perfect [Reference Gorin and JaGL69].