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Not all finitely generated groups have universal acylindrical actions

Author: Carolyn R. Abbott
Journal: Proc. Amer. Math. Soc. 144 (2016), 4151-4155
MSC (2010): Primary 20F65; Secondary 20F67
Published electronically: April 27, 2016
MathSciNet review: 3531168
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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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Carolyn R. Abbott
Affiliation: Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Drive, Madison, Wisconsin 53706

Received by editor(s): June 15, 2015
Received by editor(s) in revised form: December 17, 2015
Published electronically: April 27, 2016
Communicated by: Kevin Whyte
Article copyright: © Copyright 2016 American Mathematical Society

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