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2021 The geometry of groups containing almost normal subgroups
Alexander Margolis
Geom. Topol. 25(5): 2405-2468 (2021). DOI: 10.2140/gt.2021.25.2405

Abstract

A subgroup HG is said to be almost normal if every conjugate of H is commensurable to H. If H is almost normal, there is a well-defined quotient space GH. We show that if a group G has type Fn+1 and contains an almost normal coarse PDn subgroup H with e(GH)=, then whenever G is quasi-isometric to G it contains an almost normal subgroup H that is quasi-isometric to H. Moreover, the quotient spaces GH and GH are quasi-isometric. This generalises a theorem of Mosher, Sageev and Whyte, who prove the case in which GH is quasi-isometric to a finite-valence bushy tree. Using work of Mosher, we generalise a result of Farb and Mosher to show that for many surface group extensions ΓL, any group quasi-isometric to ΓL is virtually isomorphic to ΓL. We also prove quasi-isometric rigidity for the class of finitely presented -by-(–ended) groups.

Citation

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Alexander Margolis. "The geometry of groups containing almost normal subgroups." Geom. Topol. 25 (5) 2405 - 2468, 2021. https://doi.org/10.2140/gt.2021.25.2405

Information

Received: 22 October 2019; Revised: 3 April 2020; Accepted: 4 May 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.2405

Subjects:
Primary: 20F65
Secondary: 20E08, 20J05, 57M07

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 5 • 2021
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