The geometry of groups containing almost normal subgroups

Margolis, Alexander

doi:10.2140/gt.2021.25.2405

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The geometry of groups containing almost normal subgroups

Alexander Margolis

Geometry & Topology 25 (2021) 2405–2468

DOI: 10.2140/gt.2021.25.2405

Abstract

A subgroup $H \leq G$ 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 $G ∕ H$ . We show that if a group $G$ has type $F_{n + 1}$ and contains an almost normal coarse ${PD}_{n}$ subgroup $H$ with $e (G ∕ H) = \infty$ , 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 $G ∕ H$ and $G^{'} ∕ H^{'}$ are quasi-isometric. This generalises a theorem of Mosher, Sageev and Whyte, who prove the case in which $G ∕ H$ 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 $Γ_{L}$ , any group quasi-isometric to $Γ_{L}$ is virtually isomorphic to $Γ_{L}$ . We also prove quasi-isometric rigidity for the class of finitely presented $ℤ$ -by-( $\infty$ –ended) groups.

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Keywords

almost normal, quasi-isometry, coarse bundle

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E08, 20J05, 57M07

References

Publication

Received: 22 October 2019

Revised: 3 April 2020

Accepted: 4 May 2020

Published: 3 September 2021

Proposed: Mladen Bestvina

Seconded: David M Fisher, Anna Wienhard

Authors

Alexander Margolis
Department of Mathematics Vanderbilt University Nashville, TN United States

Volume 25, issue 5 (2021)