  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.

Group Theory