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

TL;DR

This paper proves that two-dimensional Artin groups satisfy a strong version of the Tits alternative, and for hyperbolic types, large powers of elements either commute or generate free groups, confirming Wise's conjecture.

Contribution

It establishes the Tits alternative for two-dimensional Artin groups and confirms Wise's conjecture on subgroups generated by large powers in hyperbolic cases.

Findings

Subgroups of two-dimensional Artin groups are either virtually free abelian or contain a free group.

In hyperbolic cases, large powers of elements either commute or generate free groups.

The results extend the understanding of subgroup structure in Artin groups.

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.