TITLE: Isomorphism problem among Thompson's groups
QUESTION [5 upvotes]: I am looking for a reference proving the following statement:

For every $n,m \geq 2$, the groups $T_n$ and $T_m$ are isomorphic if and only if $n=m$.

Here, $T_k$ denotes the variation of Thompson's group $T$ (compared to $F$, $T$ acts on the circle instead of the interval) where dyadic numbers are replaced with $k$-adic numbers. 
I do not focus only on the first appearance of this statement. If several proofs are available in the literature, I am also interested to know them.

REPLY [2 votes]: I would like to add a small comment to Jim Belk's detailed and clear answer.  
There is a preprint Xiaobing Sheng's work describing quasi-isometric embeddings from $T_n$ to $T$ which also provides a partial answer along the lines of Liousse's paper mentioned in J. Belk's answer.  I am not certain of the current publication status of Sheng's paper.  The relevant proposition there is:
Proposition 3.1.2. The order of a torsion element in $T_n$ is a divisor of $l(n − 1) + 1$
for some integer $l \geq 0$.