Divergence properties of the generalised Thompson groups and the braided-Thompson groups

TL;DR

This paper investigates the divergence properties of generalized Thompson groups, including Brown-Thompson and braided Thompson groups, establishing that all these groups exhibit linear divergence functions.

Contribution

It extends known divergence results from classical Thompson groups to their generalizations and braided variants, demonstrating they all have linear divergence.

Findings

Brown-Thompson groups $F_n$, $T_n$, $V_n$ have linear divergence

Braided Thompson groups $BF$, $\uhat{BF}$, $rac{BV}$ have linear divergence

Generalized Thompson groups share divergence properties with classical Thompson groups

Abstract

Golan and Sapir \cite{MR3978542} proved that the Thompson's groups $F$, $T$ and $V$ have linear divergence. In the current paper, we focus on the divergence properties of several generalisation of the Thompson's groups, we first consider the Brown-Thompson's groups $F_n$, $T_n$ and $V_n$ and found out that these groups also have linear divergence function. We then consider the braided Thompson's groups $BF$ and $\widehat{BF}$ and $\widehat{BV}$ together with the result in \cite{Kodama:2020to} we conclude that theses groups have linear divergence.