Higman–Thompson‐like groups of higher rank graph C*‐algebras

Abstract

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