#### Volume 25, issue 5 (2021)

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The geometry of groups containing almost normal subgroups

### Alexander Margolis

Geometry & Topology 25 (2021) 2405–2468
##### Abstract

A subgroup $H\le G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H\phantom{\rule{-0.17em}{0ex}}$. If $H$ is almost normal, there is a well-defined quotient space $G∕H\phantom{\rule{-0.17em}{0ex}}$. We show that if a group $G$ has type ${F}_{n+1}$ and contains an almost normal coarse ${PD}_{n}$ subgroup $H$ with $e\left(G∕H\right)=\infty$, then whenever ${G}^{\prime }$ is quasi-isometric to $G$ it contains an almost normal subgroup ${H}^{\prime }$ that is quasi-isometric to $H\phantom{\rule{-0.17em}{0ex}}$. Moreover, the quotient spaces $G∕H$ and ${G}^{\prime }∕{H}^{\prime }$ 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 ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}L}$, any group quasi-isometric to ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}L}$ is virtually isomorphic to ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}L}$. We also prove quasi-isometric rigidity for the class of finitely presented $ℤ$-by-($\infty$–ended) groups.

##### Keywords
almost normal, quasi-isometry, coarse bundle
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E08, 20J05, 57M07