# Cyclically reduced elements in Coxeter groups

## TL;DR

This paper characterizes conjugacy classes in Coxeter groups, describes cyclically reduced elements, and proves a conjecture, extending known results from finite to all Coxeter groups.

## Contribution

It provides a detailed description of conjugacy classes and cyclically reduced elements in all Coxeter groups, generalizing prior finite group results.

## Key findings

- Complete description of conjugacy classes in Coxeter groups
- Characterization of cyclically reduced elements
- Proof of Cohen's conjecture from 1994

## Abstract

Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.