Each Coxeter group W is associated to an Artin–Tits group B W , which is defined by a similar presentation that omits relations of the form s 2 = 1 for each generator s. In particular, the Artin–Tits group associated to S ~ ~ n is generated by n elements σ σ 0 , σ σ 1 , … … , σ σ n − − 1 subject to the relations σ σ i σ σ i + 1 σ σ i = σ σ i + 1 σ σ i σ σ i + 1 for i = 0 , … … , n − − 1 (and no others), where as before the indices are taken modulo n (so σ σ n = σ σ 0 ). Artin–Tits groups of Coxeter groups are conjectured to have many nice properties: for example, they are conjectured to be torsion-free, to have trivial center, to have solvable word problem, and to satisfy the K ( π π , 1 ) conjecture. These conjectures are not known to hold for all Artin–Tits groups, but in (Charney & Peifer 2003) it was shown that B S ~ ~ n has these properties. (Subsequently, they have been proved for the Artin–Tits groups associated to affine Coxeter groups.) In the case of the affine symmetric group, these proofs make use of an associated Garside structure on the Artin–Tits group.