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

TL;DR

This paper demonstrates that all finitely generated Artin-Tits groups possess a finite Garside family by introducing low elements in Coxeter groups and analyzing their properties, including finiteness and closure under certain operations.

Contribution

It introduces the concept of low elements in Coxeter groups and proves their properties, establishing the existence of finite Garside families in finitely generated Artin-Tits groups.

Findings

The set of all low elements in a Coxeter system is finite.

Low elements include the generating set S.

The set of low elements is closed under suffix and join operations.

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.