Quasi-isometric embedding from the generalised Thompson's group $T_n$ to $T$

TL;DR

This paper investigates the geometric relationships between generalized Thompson's groups, establishing the existence of quasi-isometric embeddings from $T_n$ to $T$ and proving the non-existence of embeddings in the reverse direction for certain cases.

Contribution

It demonstrates a quasi-isometric embedding from $T_n$ to $T$ for all $n \, \geq 2$, and shows that no such embeddings exist from $T_2$ to $T_n$ for $n \, \geq 3$.

Findings

Existence of quasi-isometric embedding from $T_n$ to $T$ for all $n \geq 2$.

Non-existence of embedding from $T_2$ to $T_n$ for $n \geq 3$.

Abstract

Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ 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 $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.