Acylindrically hyperbolic groups with exotic properties

TL;DR

This paper constructs a special acylindrically hyperbolic group with unique fixed point properties and unusual characteristics, expanding understanding of hyperbolic-like groups.

Contribution

It proves the existence of a common quotient for countable families of acylindrically hyperbolic groups and constructs a group with exotic fixed point and torsion properties.

Findings

Existence of a common acylindrically hyperbolic quotient for any countable family.

Construction of a group with property $FL^p$ for all $p$ and fixed points on finite-dimensional spaces.

The group is not uniformly non-amenable and has large torsion elements in its generating sets.

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 $FL^p$ 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.