Divergence, thickness and hypergraph index for general Coxeter groups

TL;DR

This paper investigates divergence and thickness in Coxeter groups, introduces a computable hypergraph index, and explores their relationships, providing bounds and conjectures supported by specific examples and topological insights.

Contribution

It introduces the hypergraph index for Coxeter systems, generalizes previous definitions, and establishes bounds on divergence and thickness related to this invariant.

Findings

Hypergraph index bounds divergence and thickness in Coxeter groups.

Finite hypergraph index implies polynomial divergence of degree at most h+1.

Constructs infinite families of Coxeter groups with the same hypergraph index.

Abstract

We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems $(W,S)$. This generalises Levcovitz's definition for the right-angled case. We prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly algebraically) thick of order at most $h$, hence has divergence bounded above by a polynomial of degree $h+1$. We conjecture that these upper bounds on the order of thickness and divergence are in fact equalities, and we prove our conjecture for certain families of Coxeter groups. These families are obtained by a new construction which, given any right-angled Coxeter group, produces infinitely many examples of non-right-angled Coxeter systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of, unexpectedly, the topology of its associated Dynkin diagram.