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 {σ _s :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 ² and σ _t ² commute whenever σ _s and σ _t commute, for s,t∈S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m _s ≥2 for each s∈S, we show that the elements {T _s =σ _s ^ms :s∈S} generate a subgroup that has a finite presentation in which the only defining relations are that T _s and T _t commute if σ _s and σ _t commute. Oblatum 21-III-2000 & 1-XII-2000?Published online: 5 March 2001