Divergence of CAT(0) Cube Complexes and Coxeter Groups

TL;DR

This paper studies the divergence properties of CAT(0) cube complexes and right-angled Coxeter groups, providing geometric conditions, classification results, and criteria for polynomial divergence degrees.

Contribution

It introduces new geometric conditions for divergence bounds, classifies right-angled Coxeter groups with quadratic divergence, and offers graph-theoretic criteria for polynomial divergence degrees.

Findings

Classified all right-angled Coxeter groups with quadratic divergence.

Showed right-angled Coxeter groups cannot have divergence between quadratic and cubic.

Provided criteria for recognizing polynomial divergence degrees in Coxeter groups.

Abstract

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. As another application, we provide an inductive graph theoretic criteria on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups which are not right-angled.