Artin-Tits groups, Coxeter groups and Shi arrangements

In this talk, I will survey how the study of the word problem - a pure abstract group problem -  in Artin-Tits group, a class of groups that generalizes the braid group, led to an unexpected link to a family of hyperplane arrangements associated to Coxeter groups - groups of mirror symmetries - called the Shi arrangements. In doing so, I will illustrate how the intuition behind this research arises from previous research on the understanding of infinite root systems and inversion sets of elements in Coxeter groups.