Shi arrangement and low element

In this talk, I will survey the notion of Garside families in relation with the word problem for Artin-Tits groups. In particular, Dehornoy, Dyer and I proved that there exist a finite Garside family in any Artin-Tits group G. Then I will discuss how this family is (suprisingly) related to a well-known finite hyperplane sub-arrangements, the Shi arrangement, of the reflection arrangement of the Coxeter group associated to G.