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 $n\geq 1$ such that $a^n$ and $b^n$ either commute or generate a non-abelian free subgroup.