The geometry of groups containing almost normal subgroups

Abstract

A subgroup $H\le G$ 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 $e\left(G∕H\right)=\infty$, then whenever $G^{\prime }$ is quasi-isometric to $G$ it contains an almost normal subgroup $H^{\prime }$ that is quasi-isometric to $H$. 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 ${\mathrm{\Gamma }}_L$, any group quasi-isometric to ${\mathrm{\Gamma }}_L$ is virtually isomorphic to ${\mathrm{\Gamma }}_L$. We also prove quasi-isometric rigidity for the class of finitely presented $\mathbb{Z}$-by-($\infty$–ended) groups.