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 {sigma (s) : s epsilon S}. Tits conjectured that the only relations in A amongst the squares of t he generators are consequences of the obvious ones, namely that sigma (2)(s ) and sigma (2)(t) commute whenever sigma (s) and sigma (t) commute, for s, t epsilon S. In this paper we prove Tits' conjecture for all Artin groups. In fact, given a number m(s) greater than or equal to 2 for each s epsilon S, we show that the elements {T-S = sigma (ms)(S) : s epsilon S} generate a subgroup that has a finite presentation in which the only defining relati ons are that T-s and T-t commute if sigma (s) and sigma (t) commute.