Irrational-slope versions of Thompson's groups $T$ and $V$

Abstract

In this paper we consider the $T$- and $V$- versions, $T_{\tau }$ and $V_{\tau }$ , of the irrational slope Thompson group $F_{\tau }$ considered in [3]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for $T$ and $V$. We also show that $T_{\tau }$ and $V_{\tau }$ have index-2 normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-2 subgroups are shown to be simple.

21 pages, 7 figures