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

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.