The Tits alternative for two-dimensional Artin groups and Wise's power alternative

Abstract

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2.When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a,b of a two-dimensional Artin group of hyperbolic type, there exists an integer n≥1 such that an and bn either commute or generate a non-abelian free subgroup.