Garside families in Artin–Tits monoids and low elements in Coxeter groups

Abstract

We show that every finitely generated Artin–Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W,S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.