Divergence function of the braided Thompson group

TL;DR

This paper proves that the braided Thompson group has a linear divergence function, implying that its asymptotic cones do not have cut-points, which informs its large-scale geometric structure.

Contribution

It establishes the linear divergence property of the braided Thompson group, a new geometric insight into its asymptotic behavior.

Findings

Braided Thompson group has linear divergence.

Asymptotic cones of BV lack cut-points.

Provides geometric classification of BV.

Abstract

We prove that the braided Thompson group $BV$ has a linear divergence function. By the work of Dru\c{t}u, Mozes, and Sapir, this leads none of asymptotic cones of $BV$ has a cut-point.