Conjugacy classes and straight elements in Coxeter groups

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W0 of W, any two cyclically reduced expressions of conjugate elements of W0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a very simple description of conjugacy classes in W0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w∈W for which ℓ(wn)=nℓ(w) for any n∈Z. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight.