Examples of cubulable groups with fixed-point properties

Abstract

For every $n\ge 1$, let $({\mathrm{FW}}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups satisfying $({\mathrm{FW}}_n)$ but with good cubical properties in higher dimensions. First, we prove that, for a finitely generated group $G$ with no non-abelian free subgroup, $G$ satisfies $({\mathrm{FW}}_n)$ if and only if no subgroup $H\le G$ of index $\le n$ can be mapped to ${\mathbb{D}}_{\infty }$ with an infinite image. For instance, the affine Coxeter group ${\tilde{A}}_n$ satisfies $({\mathrm{FW}}_n)$ but not $({\mathrm{FW}}_{n+1})$. In another direction, we investigate virtually graph products of finite groups. As an application of our constructions, we find explicit examples, for every $n\ge 1$, of acylindrically hyperbolic groups that are cocompactly cubulable but satisfy $({\mathrm{FW}}_n)$. Several conjectures and open questions are included.

26 pages. Version 2: Theorem 1.1 generalised from virtually abelian groups to groups with no non-abelian free subgroups