Higman-Thompson Like Groups of Higher Rank Graph C*-Algebras

Let $\mathrm{\Lambda }$ be a row-finite and source-free higher rank graph with finitely many vertices. In this paper, we define the Higman-Thompson like group $\text{Lht}$ of the graph C*-algebra ${\mathcal{O}}_{\mathrm{\Lambda }}$ to be a special subgroup of the unitary group in ${\text{O}}_{\mathrm{\Lambda }}$. It is shown that $\text{Lht}$ is closely related to the topological full groups of the groupoid associated with $\mathrm{\Lambda }$. Some properties of $\text{Lht}$ are also investigated. We show that its commutator group $\text{DLht}$ is simple and that $\text{DLht}$ has only one nontrivial uniformly recurrent subgroup if $\mathrm{\Lambda }$ is aperiodic and strongly connected. Furthermore, if $\mathrm{\Lambda }$ is single-vertex, then we prove that $\text{Lht}$ is C*-simple and also provide an explicit description on the stabilizer uniformly recurrent subgroup of $\text{Lht}$ under a natural action on the infinite path space of $\mathrm{\Lambda }$.