Reflection length in non-affine Coxeter groups

TL;DR

This paper proves that in any non-affine Coxeter group, the reflection length of elements can grow arbitrarily large, confirming a conjecture and introducing new tools involving hyperbolic quotients.

Contribution

It establishes the unboundedness of reflection length in non-affine Coxeter groups and constructs hyperbolic quotients as a novel approach.

Findings

Reflection length is unbounded in non-affine Coxeter groups

Construction of word-hyperbolic quotients for minimal non-affine Coxeter groups

Confirmation of McCammond and Petersen's conjecture

Abstract

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the construction of word-hyperbolic quotients of all minimal non-affine Coxeter groups might be of independent interest.