Topological full groups of $C^*$-algebras arising from $\beta$-expansions

We will introduce a family $\Gamma_\beta, 1 < \beta \in {\mathbb{R}}$ of infinite non-amenable discrete groups as an interpolation of the Higman-Thompson groups $V_n, 1 < n \in {\mathbb{N}}$ by using the topological full groups of the groupoids defined by $\beta$-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras. The groups $\Gamma_\beta, 1 < \beta \in {\mathbb{R}}$ are realized as groups of piecewise linear functions on $[0,1]$ if the $\beta$-expansion of $1$ is finite or ultimately periodic. We also classify the groups $\Gamma_\beta, 1 < \beta \in {\mathbb{R}}$ by the number theoretical property of $\beta$.