No 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.

arXiv comment13 pages. v2: introduction slightly rewritten, with added references. v3: final version. To appear in Proceedings of the AMS