No quasi-isometric rigidity for proper actions on CAT(0) cube complexes

TL;DR

This paper demonstrates that quasi-isometric groups can have fundamentally different actions on CAT(0) cube complexes, showing no quasi-isometric rigidity in this context.

Contribution

It provides the first examples of groups with the same large-scale geometry but different proper actions on median graphs, answering a longstanding question.

Findings

Existence of groups acting properly on median graphs with non-quasi-isometric counterparts

Examples are central extensions of cubulated groups

Shows lack of quasi-isometric rigidity for proper actions on CAT(0) cube complexes

Abstract

We exhibit a variety of groups that act properly and even cocompactly on median graphs (a.k.a. one-skeletons of CAT(0) cube complexes), with quasi-isometric groups that do not admit any proper action on a median graph. This answes a question of Niblo, Sageev and Wise. Our examples are all quasi-isometrically trivial central extensions of certain cubulated groups.