Nombre de classes de conjugaison d'\'el\'ements d'ordre fini dans les groupes de Brown-Thompson

TL;DR

This paper determines the number of conjugacy classes of finite order elements in Brown-Thompson groups, extending previous results on Thompson groups and establishing conditions for isomorphisms.

Contribution

It generalizes Matucci's results to Brown-Thompson groups, providing explicit formulas for conjugacy classes and proving non-isomorphism with Thompson group T.

Findings

Exact count of conjugacy classes of elements of order q in T_{r,m}

Conditions under which elements of finite order exist in T_{r,m}

Thompson group T is not isomorphic to T_{r,m} for certain parameters

Abstract

We extend a result of Matucci on the number of conjugacy classes of finite order elements in the Thompson group $T$. According to Liousse, if $ gcd(m-1,q)$ is not a divisor of $r$ then there does not exist element of order $q$ in the Brown-Thompson group $T_{r,m}$. We show that if $ gcd(m-1,q)$ is a divisor of $r$ then there are exactly $\varphi(q). gcd(m-1,q)$ conjugacy classes of elements of order $q$ in $T_{r,m}$, where $\varphi$ is the Euler function phi. As a corollary, we obtain that the Thompson group $T$ is isomorphic to none of the groups $T_{r,m}$, for $m\not=2$ and any morphism from $T$ into $T_{r,m}$, with $m\not=2$ and $r\not= 0$ $mod \ (m-1)$, is trivial.