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2018 Detecting a subclass of torsion-generated groups
Emily Stark
Algebr. Geom. Topol. 18(7): 4037-4068 (2018). DOI: 10.2140/agt.2018.18.4037

Abstract

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurable to a group generated by finite-order elements. Consequently, the subclass of groups in this class generated by finite-order elements is not quasi-isometrically rigid. We provide necessary conditions for two groups in this class to be abstractly commensurable. We use these conditions to prove there are infinitely many abstract commensurability classes within each quasi-isometry class of this class that contains a group generated by finite-order elements.

Citation

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Emily Stark. "Detecting a subclass of torsion-generated groups." Algebr. Geom. Topol. 18 (7) 4037 - 4068, 2018. https://doi.org/10.2140/agt.2018.18.4037

Information

Received: 19 November 2017; Revised: 31 July 2018; Accepted: 23 August 2018; Published: 2018
First available in Project Euclid: 18 December 2018

zbMATH: 07006384
MathSciNet: MR3892238
Digital Object Identifier: 10.2140/agt.2018.18.4037

Subjects:
Primary: 20F65
Secondary: 20E08, 20F55, 57M07, 57M20

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 7 • 2018
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