Minimality and effectiveness of the groupoid associated to a self-similar ultragraph
Hossein Larki, Najmeh Rajabzadeh-Hasiri
TL;DR
This work analyzes the minimality and effectiveness of the tight groupoid $\mathcal{G}_{\mathrm{tight}}(\mathcal{S}_{G,\mathcal{U}})$ arising from a self-similar ultragraph $(G,\mathcal{U},\varphi)$ and connects these dynamical properties to simplicity results for the associated $C^*$-algebra $\mathcal{O}_{G,\mathcal{U}}$. By leveraging the Exel–Pardo framework, the authors identify a drive-by criterion called $G$-cofinality that characterizes minimality, and a topological-freeness criterion (Condition $(*)$) together with entrances for every $G$-cycle that yields effectiveness. In the special case of a trivial 1-cocycle, they obtain a crossed-product form $\mathcal{O}_{G,\mathcal{U}} \cong C^*(\mathcal{U}) \rtimes_{\eta} G$ and prove simplicity under amenability of $G$ and pseudo-freeness, together with the condition $(*)$, while establishing a unitary representation of $G$ in the multiplier algebra. These results provide a robust groupoid–inverse-semigroup route to understanding when self-similar ultragraph algebras are simple and nuclear, with explicit crossed-product realizations in the key case.
Abstract
The notion of a self-similar ultragraph $(G,\mathcal{U},\varphi)$ and its $C^*$-algebra $\mathcal{O}_{G,\mathcal{U}}$ were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such $C^*$-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph $(G,\mathcal{U},\varphi)$. In particular, we obtain a result for simplicity of the $C^*$-algebras $\mathcal{O}_{G,\mathcal{U}}$ in a certain case.