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 $P{D}_n$ subgroup $H$ with $e(G/H)=\infty$, 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-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-Mosher to show that for many surface group extensions ${\Gamma }_L$, any group quasi-isometric to ${\Gamma }_L$ is virtually isomorphic to ${\Gamma }_L$. We also prove quasi-isometric rigidity for the class of finitely presented $\mathbb{Z}$-by-($\infty$ ended) groups.

48 pages

References in corpus (2)

• Commensurating HNN-extensions: non-positive curvature and biautomaticity
• Fiber respecting quasi-isometries of surface group extensions

Cited by in corpus (3)

• Quasi-isometric rigidity of subgroups and Filtered ends
• Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups
• Hyperbolically embedded subgroups and quasi-isometries of pairs