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

arXiv comment29 pages, 10 figures, Major revisions on Lemma 3.6, Corollary 3.7, the proof of Proposition 3.9, lemma 4.9, lemma 4.10, lemma 4.12, lemma 4.17 and proposition 4.19 made. Minor errors corrected. Suggestions and comments are welcome!