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

TL;DR

This paper introduces a Higman-Thompson like group associated with higher rank graph C*-algebras, exploring its properties, simplicity, and subgroup structure, linking it to topological full groups of associated groupoids.

Contribution

It defines a new Higman-Thompson like group for higher rank graph C*-algebras and investigates its algebraic and topological properties, including simplicity and subgroup classifications.

Findings

The commutator subgroup of the group is simple.

The group has a unique nontrivial uniformly recurrent subgroup under certain conditions.

The group is C*-simple when the graph has a single vertex.

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 $\Lambda$.