The Tits Alternative for two-dimensional Artin groups and Wise's Power Alternative

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.