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 a n and b n either commute or generate a non -ab elian free subgroup. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).