Quasi-isometric groups with no common model geometry

TL;DR

This paper investigates the geometric properties of groups derived from simple surface amalgams, showing that quasi-isometric groups may not share a common model geometry, highlighting limitations in geometric group classification.

Contribution

It proves that fundamental groups of simple surface amalgams acting on the same space are commensurable and demonstrates the existence of infinitely many quasi-isometric groups with different model geometries.

Findings

Groups are commensurable if they act on the same space

Existence of infinitely many quasi-isometric groups with different geometries

Limitations in classifying groups by their model geometries

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 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.