Two-generator subgroups of the pure braid group

TL;DR

This paper proves that in the pure braid group, any two elements either commute or generate a free subgroup, using 3-manifold theory and group actions on trees.

Contribution

It resolves a question by Luis Paris by establishing a clear dichotomy for pairs of elements in the pure braid group.

Findings

Any two elements commute or generate a free group

The proof involves 3-manifold theory and group actions on trees

Settles a longstanding question in the theory of pure braid groups

Abstract

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.