# The geometry of groups containing almost normal subgroups

## TL;DR

This paper explores the geometric properties of groups with almost normal subgroups, demonstrating quasi-isometric invariance and rigidity results, extending previous theorems to broader classes of groups and quotient spaces.

## Contribution

It generalizes existing theorems on almost normal subgroups and quasi-isometric rigidity, applying to a wider class of groups and quotient space structures.

## Key findings

- Quasi-isometric invariance of almost normal subgroups with certain properties.
- Extension of Mosher-Sageev-Whyte theorem to broader quotient spaces.
- Quasi-isometric rigidity for specific classes of finitely presented groups.

## 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-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.