In this talk I will discuss a relative to Thompson's group $F$, the group $F_\tau,$ which is the group of piecewise linear homeomorphisms of $[0,1]$ with breakpoints in $\mathbb{Z}[\tau]$ and slopes powers of $\tau,$ where $\tau = \frac{\sqrt5 -1}{2}$ is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type $\operatorname{F}_\infty.$

Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to show that the commutator subgroup is simple and give a unique normal form for its elements. The surprising feature is that the $T$- and $V$-versions of these groups are not simple, however, but are also of type $\operatorname{F}_\infty.$ This is joint work with J. Burillo and L. Reeves.

If time permits I will give an outline of the difficulties coming up when considering other algebraic numbers as slopes and breakpoints.