Coarse obstructions to cocompact cubulation

TL;DR

This paper develops geometric methods to identify obstructions to cocompact cubulation of groups, revealing new limitations and examples in the context of CAT(0) cube complexes and coarse median spaces.

Contribution

It introduces novel geometric bounds and obstructions that prevent certain groups from being cocompactly cubulated, including new examples and counterexamples.

Findings

Many free-by-cyclic groups cannot be cocompactly cubulated.

A CAT(0), $C(6)$, virtually special group is not quasiisometric to any CAT(0) cube complex.

First example of a $C(6)$ group that cannot be cocompactly cubulated.

Abstract

We provide geometric methods to give bounds on the large-scale dimension of CAT(0) cube complexes quasiisometric to a given group $G$. In situations where these bounds conflict we obtain obstructions to $G$ being cocompactly cubulated. More strongly, the obstructions prevent $G$ from being a coarse median space. As applications, we show that many free-by-cyclic groups cannot be cocompactly cubulated, even virtually, and prove that any tubular group with a coarse median is virtually compact special. We also exhibit a group that is CAT(0), $C(6)$, and virtually special, yet is not quasiisometric to any CAT(0) cube complex. This is the first example of a $C(6)$ group that cannot be cocompactly cubulated, resolving a question of Jankiewicz and partially answering a question of Wise.