Subgroup growth of right-angled Artin and Coxeter groups

TL;DR

This paper investigates how the number of finite index subgroups in right-angled Artin groups grows, showing it depends on the graph's independence number, and explores similar properties for Coxeter groups.

Contribution

It precisely determines the subgroup growth rate for right-angled Artin groups and proposes a conjecture for Coxeter groups, with partial proofs.

Findings

Growth rate depends only on the independence number of the defining graph.

Established the growth rate formula for right-angled Artin groups.

Proposed and proved a conjecture for right-angled Coxeter groups in limited cases.

Abstract

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.