Topological full groups of $C^{\ast }$-algebras arising from $\beta$-expansions

Abstract

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 $\beta$.

27 pages

References in corpus (2)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• Homology and topological full groups of etale groupoids on totally disconnected spaces