Quasi‐isometric groups with no common model geometry

Abstract

A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove that if two fundamental groups of simple surface amalgams act properly and cocompactly by isometries on the same proper geodesic metric space, then the groups are commensurable. Consequently, there are infinitely many fundamental groups of simple surface amalgams that are quasi‐isometric, but which do not act properly and cocompactly on the same proper geodesic metric space.

Action rigidity for free products of hyperbolic manifold groups

“…The first step is geometric; we show that geometric actions of two infinite-ended non-free hyperbolic groups on an arbitrary common model space can be promoted to geometric actions on a model space with more structure. This strategy to prove action rigidity was also employed by Mosher-Sageev-Whyte [38] for the virtually free groups defined above and by the authors [47] for the class of simple surface amalgams. The second step is topological; we prove a generalization of Leighton's graph covering theorem [34], following the methods developed by Woodhouse [53], and Shepherd and Gardam-Woodhouse [45].…”

“…geometry -the proof exploits the fact that any proper, minimal action of G p on a simplicial tree must be the natural action on the p-regular tree. A class of groups called simple surface amalgams gives examples of torsionfree hyperbolic groups that are quasi-isometric but do not have a common model geometry, as shown by the authors [47].…”

Hyperbolic Groups That Are Not Commensurably Co-Hopfian

“…Let C k denote the set of groups π 1 (X) such that X is the simple surface amalgam obtained from k surfaces with boundary. Simple surface amalgams have been studied in [Mal10,Sta17,DST18,SW18].…”