# A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

## TL;DR

This paper introduces the hypergraph index, a new quasi-isometry invariant for 2D right-angled Coxeter groups, which helps classify these groups and bounds their geometric properties, providing new insights into their thickness and divergence.

## Contribution

The paper defines the hypergraph index for right-angled Coxeter groups, enabling classification into quasi-isometry classes and establishing bounds on their geometric properties.

## Key findings

- Hypergraph index partitions groups into infinitely many classes.
- Hypergraph index can be computed from the defining graph.
- Examples of groups with different thickness and algebraic thickness orders.

## Abstract

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n>1, we give examples of right-angled Coxeter groups which are thick of order n, yet are algebraically thick of order strictly larger than n, answering a question of Behrstock-Drutu-Mosher.