Periodic 2-graphs arising from subshifts

TL;DR

This paper introduces a new family of periodic 2-graphs called domino graphs, explores their combinatorial properties, and analyzes the structure of their associated $C^*$-algebras, expanding the understanding of higher-rank graph algebras.

Contribution

The paper constructs and analyzes a new class of periodic 2-graphs, called domino graphs, and provides a structure theorem for their $C^*$-algebras, linking combinatorics and operator algebras.

Findings

Introduction of domino graphs as periodic 2-graphs

Analysis of combinatorial structure of domino graphs

Proven structure theorem for the $C^*$-algebras of domino graphs

Abstract

Higher-rank graphs were introduced by Kumjian and Pask to provide models for higher-rank Cuntz-Krieger algebras. In a previous paper, we constructed 2-graphs whose path spaces are rank-two subshifts of finite type, and showed that this construction yields aperiodic 2-graphs whose $C^*$-algebras are simple and are not ordinary graph algebras. Here we show that the construction also gives a family of periodic 2-graphs which we call \emph{domino graphs}. We investigate the combinatorial structure of domino graphs, finding interesting points of contact with the existing combinatorial literature, and prove a structure theorem for the $C^*$-algebras of domino graphs.