The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

Let A be an Artin group with standard generating set {σs : s ∈ S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σ s and σ t commute whenever σs and σt commute, for s, t ∈ S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number ms ≥ 2 for each s ∈ S, we show that the elements {Ts = σs s : s ∈ S} generate a subgroup that has a finite presentation in which the only defining relations are that Ts and Tt commute if σs and σt commute.