Quasi-isometric Embeddings from Generalised Thompson’s Groups to Thompson’s Group T

Abstract

Brown has defined generalised Thompson’s groups $F_n$, $T_n$, $V_n$ where $n$ is an integer at least $2$ and Thompson’s groups $F=F_2$, $T=T_2$, $V=V_2$ in the 80’s. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n\ge 2$ and no embeddings from $T_2$ to $T_n$ for $n\ge 3$.