Bounding reflection length in an affine Coxeter group

TL;DR

This paper proves that in affine Coxeter groups, the reflection length of any element is uniformly bounded above by twice the dimension, establishing an optimal bound and conjecturing this property is unique to spherical and affine types.

Contribution

It establishes a tight upper bound of 2n for reflection length in affine Coxeter groups and conjectures this bound characterizes these groups among all Coxeter groups.

Findings

Reflection length in affine Coxeter groups is bounded by 2n.

The bound of 2n is proven to be optimal.

Conjecture that only spherical and affine Coxeter groups have such bounds.

Abstract

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on $\R^n$ is bounded above by $2n$ and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.