Not all finitely generated groups have universal acylindrical actions

TL;DR

The paper demonstrates that not all finitely generated groups admit a universal acylindrical action where all generalized loxodromic elements are loxodromic, providing a counterexample to a question posed by Osin.

Contribution

It shows that there exist finitely generated groups without a universal acylindrical action, answering Osin's question negatively using Dunwoody's inaccessible group as a counterexample.

Findings

Counterexample to Osin's question using Dunwoody's group

Not all finitely generated groups have universal acylindrical actions

Some groups lack a hyperbolic action capturing all generalized loxodromic elements

Abstract

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $\operatorname{Out}(\mathbb F_n)$ for $n\geq 2$. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. We answer this question in the negative, using Dunwoody's example of an inaccessible group as a counterexample.