The geometry of conjugation in affine Coxeter groups

We develop precise geometric descriptions of the conjugacy class [Formula: see text] and coconjugation set [Formula: see text] for all elements [Formula: see text] of any affine Coxeter group [Formula: see text]. The centralizer of x in [Formula: see text] is the special case [Formula: see text]. The key structure in our description of the conjugacy class [Formula: see text] is the mod-set [Formula: see text], where w is the finite part of x and [Formula: see text] is the coroot lattice. The set [Formula: see text] is then described by [Formula: see text] together with the fix-set of [Formula: see text], where [Formula: see text] is the finite part of [Formula: see text]. For any element w of the associated finite Weyl group W, the mod-set of w is contained in the classical move-set [Formula: see text]. We prove that the rank of [Formula: see text] equals the dimension of [Formula: see text] and investigate type-by-type the surprisingly subtle structure of [Formula: see text].