# Amenability and acyclicity in bounded cohomology

### Marco Moraschini

Università di Bologna, Italy### George Raptis

Universität Regensburg, Germany

## Abstract

Johnson’s characterization of amenable groups states that a discrete group $\Gamma$ is amenable if and only if $H_b^{n \geq 1}(\Gamma; V) = 0$ for all dual normed $\mathbb{R}[\Gamma]$-modules $V$. In this paper, we extend the previous result to homomorphisms by proving the converse of the *mapping theorem*: a surjective group homomorphism $\phi \colon \Gamma {\to} K$ has amenable kernel $H$ if and only if the induced inflation map $H^\bullet_b(K; V^H) {\to} H^\bullet_b(\Gamma; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[\Gamma]$-module $V$. In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms $\phi \colon \Gamma \to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for *all* Banach $\Gamma$-modules. Finally, we also prove a characterization of the (larger) class of *boundedly acyclic* homomorphisms, that is, the class of group homomorphisms $\phi \colon \Gamma \to K$ for which the restriction maps in bounded cohomology $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \phi^{-1}V)$ are isomorphisms for a suitable family of dual normed $\mathbb{R}[K]$-modules $V$ including the trivial $\mathbb{R}[K]$-module $\mathbb{R}$. We then extend the first and third results to topological spaces and obtain characterizations of *amenable* maps and *boundedly acyclic* maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.

## Cite this article

Marco Moraschini, George Raptis, Amenability and acyclicity in bounded cohomology. Rev. Mat. Iberoam. (2023),

DOI 10.4171/RMI/1406