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

摘要

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~(m_s):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.