Quasi-positivity and recognition of products of conjugacy classes in free groups

TL;DR

This paper introduces an algorithm to recognize quasi-positive elements in free groups, which is significant for understanding braid groups and their geometric properties, using spherical cancellation diagrams for validation.

Contribution

It presents a novel algorithm for identifying quasi-positive elements in free groups, supported by spherical cancellation diagrams for correctness and runtime analysis.

Findings

Algorithm accurately recognizes quasi-positive elements.

Spherical cancellation diagrams validate the algorithm.

Provides worst-case runtime complexity.

Abstract

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with the geometrically defined class of $\mathbb{C}$-transverse links. We describe an algorithm that recognizes whether or not an element of a free group is quasi-positive with respect to a basis. Spherical cancellation diagrams over free groups are used to establish the validity of the algorithm and to determine the worst-case runtime.