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

In this paper we consider the $T$- and $V$- versions, $T_τ$ and $V_τ$ , of the irrational slope Thompson group $F_τ$ 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 $T_τ$ and $V_τ$ have index-2 normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-2 subgroups are shown to be simple.

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