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The geometry of groups containing almost normal subgroups

Alexander Margolis

Geometry & Topology 25 (2021) 2405–2468

A subgroup H 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 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.

almost normal, quasi-isometry, coarse bundle
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E08, 20J05, 57M07
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
Alexander Margolis
Department of Mathematics
Vanderbilt University
Nashville, TN
United States