Structure of conjugacy classes in Coxeter groups

This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let $(W,S)$ be a Coxeter system. A cyclic shift of an element $w\in W$ is a conjugate of $w$ of the form $sws$ for some simple reflection $s\in S$ such that $\ell_S(sws)\leq\ell_S(w)$. The cyclic shift class of $w$ is then the set of elements of $W$ that can be obtained from $w$ by a sequence of cyclic shifts. Given a subset $K\subseteq S$ such that $W_K:=\langle K\rangle\subseteq W$ is finite, we also call two elements $w,w'\in W$ $K$-conjugate if $w,w'$ normalise $W_K$ and $w'=w_0(K)ww_0(K)$, where $w_0(K)$ is the longest element of $W_K$. Let $\mathcal O$ be a conjugacy class in $W$, and let $\mathcal O^{\min}$ be the set of elements of minimal length in $\mathcal O$. Then $\mathcal O^{\min}$ is the disjoint union of finitely many cyclic shift classes $C_1,\dots,C_k$. We define the structural conjugation graph associated to $\mathcal O$ to be the graph with vertices $C_1,\dots,C_k$, and with an edge between distinct vertices $C_i,C_j$ if they contain representatives $u\in C_i$ and $v\in C_j$ such that $u,v$ are $K$-conjugate for some $K\subseteq S$. In this paper, we compute explicitely the structural conjugation graph associated to any (possibly twisted) conjugacy class in $W$, and show in particular that it is connected (that is, any two conjugate elements of $W$ differ only by a sequence of cyclic shifts and $K$-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element $w\in W$, as well as the existence of natural decompositions of $w$ as a product of a "torsion part" and of a "straight part", with useful properties.