A subgroup is said to be almost normal if every conjugate of is commensurable to . If is almost normal, there is a well-defined quotient space . We show that if a group has type and contains an almost normal coarse subgroup with , then whenever is quasi-isometric to it contains an almost normal subgroup that is quasi-isometric to . Moreover, the quotient spaces and are quasi-isometric. This generalises a theorem of Mosher, Sageev and Whyte, who prove the case in which 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 , any group quasi-isometric to is virtually isomorphic to . We also prove quasi-isometric rigidity for the class of finitely presented -by-(–ended) groups.
"The geometry of groups containing almost normal subgroups." Geom. Topol. 25 (5) 2405 - 2468, 2021. https://doi.org/10.2140/gt.2021.25.2405