Not all finitely generated groups have universal acylindrical actions

Abbott, Carolyn

doi:10.1090/proc/13101

Not all finitely generated groups have universal acylindrical actions
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by Carolyn R. Abbott PDF

Proc. Amer. Math. Soc. 144 (2016), 4151-4155 Request permission

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.

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