Amenability and Acyclicity in Bounded Cohomology Theory

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 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 homomorphisms $\phi \colon \Gamma \to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed $\mathbb{R}[\Gamma]$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic 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 every dual normed $\mathbb{R}[K]$-module $V$. We then extend the first and the third situations to 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.