Weak hyperbolicity of cube complexes and quasi-arboreal groups

TL;DR

This paper shows that certain graphs associated with CAT(0) cube complexes are tree-like, leading to new insights into the hyperbolic properties and asymptotic dimensions of groups acting on these complexes.

Contribution

It establishes the quasi-isometry of the intersection graph to a tree and generalizes finite asymptotic dimension results for groups acting on cube complexes.

Findings

The intersection graph is quasi-isometric to a tree.

Groups acting on finite-dimensional cube complexes have finite asymptotic dimension.

A cubical flat plane theorem is proved using contact graph techniques.

Abstract

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of $\Gamma$.