Characterizing divergence and thickness in right-angled Coxeter groups

TL;DR

This paper classifies divergence functions in right-angled Coxeter groups, showing they are either polynomial, exponential, or infinite, and relates divergence to the group's thickness and a new hypergraph index.

Contribution

It provides a complete classification of divergence functions in RACGs and introduces the hypergraph index to compute divergence from the defining graph.

Findings

Divergence in RACGs is either polynomial, exponential, or infinite.

Strong thickness of order k corresponds to polynomial divergence of degree k+1.

Hypergraph index allows easy computation of divergence from the defining graph.

Abstract

We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential or infinite. We prove that a RACG is strongly thick of order k if and only if its divergence function is a polynomial of degree k+1. Moreover, we show that the exact divergence function of a RACG can easily be computed from its defining graph by an invariant we call the hypergraph index.