Failure of quasi-isometric rigidity for infinite-ended groups

TL;DR

This paper demonstrates that certain infinite-ended groups with specific subgroup properties are not quasi-isometrically rigid, contrasting with the rigidity of residually-finite multi-ended hyperbolic groups which are virtually free.

Contribution

It establishes the failure of quasi-isometric rigidity for a class of infinite-ended groups and characterizes when residually-finite multi-ended hyperbolic groups are rigid.

Findings

Infinite-ended groups with certain subgroup properties are not quasi-isometrically rigid

Residually-finite multi-ended hyperbolic groups are quasi-isometrically rigid iff virtually free

Adapts Whyte's argument for commensurability of free products of hyperbolic surface groups

Abstract

We prove that an infinite-ended group whose one-ended factors have finite-index subgroups and are in a family of groups with a nonzero multiplicative invariant is not quasi-isometrically rigid. Combining this result with work of the first author proves that a residually-finite multi-ended hyperbolic group is quasi-isometrically rigid if and only if it is virtually free. The proof adapts an argument of Whyte for commensurability of free products of closed hyperbolic surface groups.