Let $G$ be a group quasi-isometric to the fundamental group of a genus 2 surface group $H$. It is well known that $G$ is quasi-isometrically rigid, i.e. $G$ and $H$ are virtually isomorphic. Does the stronger property, that $G$ and $H$ are commensurable, also hold?


If so is there a reference for this?



Quasi-isometric rigidity of surface groups and commensurability
Yes (I assume that by "virtually isomorphic" you mean commensurable modulo finite kernels, which is a nonstandard misleading use of "virtually"). This is because surface groups have Serre's property D$\_2$ meaning that each 2-cohomology class (in a finite abelian group with trivial action) it trivial on some finite index subgroup.


(Also a side remark: the known result on QI rigidity is stronger, since it says that every group QI to this surface group is, modulo a finite kernel, isomorphic to a cocompact lattice in the isometry group of the hyperbolic plane. I.e., there's a structural statement which does not require passing to a finite index subgroup.)


