The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group John Crisp and Luis Paris

Let A be an Artin group with standard generating set Σ. Tits conjectured that the only relations in A amongst the squares of the generators are the obvious ones, namely that σ2 and τ2 commute whenever σ and τ commute, for σ, τ ∈ Σ. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m(σ) ≥ 2 for each σ ∈ Σ, we show that the elements {T (σ) = σm(σ) : σ ∈ Σ} generate a subgroup in which the only relations are that T (σ) and T (τ) commute if σ and τ commute. AMS Subject Classification: Primary 20F36; Secondary 57N05