Acylindrically hyperbolic groups with exotic properties

Abstract

We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group $Q$ with strong fixed point properties: $Q$ has property $F{L}^p$ for all $p\in [1,+\infty )$, and every action of $Q$ on a finite dimensional contractible topological space has a fixed point. In addition, $Q$ has other properties which are rather unusual for groups exhibiting "hyperbolic-like" behaviour. E.g., $Q$ is not uniformly non-amenable and has finite generating sets with arbitrary large balls consisting of torsion elements.

References in corpus (4)

• Random groups, random graphs and eigenvalues of p-Laplacians
• Groups acting acylindrically on hyperbolic spaces
• Erratum to the paper "Acylindrical hyperbolicity of groups acting on trees"
• Gromov's random monsters do not act non-elementarily on hyperbolic spaces

Cited by in corpus (5)

• Groups acting acylindrically on hyperbolic spaces
• A topological zero-one law and elementary equivalence of finitely generated groups
• Cohomology of group theoretic Dehn fillings II
• Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-superrigidity
• Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces