The geometry of groups containing almost normal subgroups

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ commensurable $H$. If normal, there a well-defined quotient space $G/H$. We show that group $G$ has type $F_{n+1}$ and contains an coarse $PD_n$ with $e(G/H)=\infty$, then whenever $G'$ quasi-isometric $G$, it $H'$ Moreover, the spaces $G/H$ $G'/H'$ are quasi-isometric. This generalises theorem Mosher-Sageev-Whyte, who prove case in which finite valence bushy tree. Using work Mosher, we generalise result Farb-Mosher for many surface extensions $\Gamma_L$, any $\Gamma_L$ virtually isomorphic $\Gamma_L$. also rigidity class finitely presented $\mathbb{Z}$-by-($\infty$ ended) groups.