Metric properties of the braided Thompson's groups

TL;DR

This paper investigates the metric properties of braided Thompson's groups, providing bounds on word length based on diagram features, thus enhancing understanding of their geometric structure.

Contribution

It offers new bounds for word length in braided Thompson's groups, connecting algebraic and diagrammatic representations.

Findings

Derived upper and lower bounds for word length

Connected diagram features to algebraic complexity

Enhanced understanding of the groups' metric geometry

Abstract

Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.