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

Abstract

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