Self-similar group actions on ultragraphs and associated $C^*$-algebras

TL;DR

This paper generalizes the concept of self-similar graphs to ultragraphs, developing new $C^*$-algebras through inverse semigroup and groupoid models, expanding the framework for analyzing complex algebraic structures.

Contribution

It introduces self-similar group actions on ultragraphs and constructs their $C^*$-algebras using inverse semigroup and tight groupoid approaches, extending prior graph-based models.

Findings

Defined self-similar actions on ultragraphs

Constructed associated $C^*$-algebras

Connected models to inverse semigroup and groupoid theory

Abstract

As a generalization of the Exel-Pardo's notion of self-similar graph, we introduce self-similar group actions on ultragraphs and their $C^*$-algebras. We then approach to the $C^*$-algebras by inverse semigroup and tight groupoid models.