# Orbit equivalence of higher-rank graphs

## TL;DR

This paper investigates various forms of orbit equivalence and conjugacy in higher-rank graphs, establishing connections with groupoid isomorphisms and algebraic structures like $C^*$-algebras.

## Contribution

It characterizes continuous orbit equivalence, conjugacy, and their relations to algebraic invariants for higher-rank graphs.

## Key findings

- Orbit equivalence preserves boundary path periodicity.
- Isomorphism of boundary path groupoids characterizes orbit equivalence.
- Relations between graph conjugacies and algebraic structures are established.

## Abstract

We study the notions of continuous orbit equivalence and eventual one-sided conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of row-finite higher-rank graphs with finitely many vertices and no sinks or sources. We show that there is a continuous orbit equivalence between two finitely-aligned higher-rank graphs that preserves the periodicity of boundary paths if and only if the boundary path groupoids are isomorphic, and characterise continuous orbit equivalence, eventual one-sided conjugacy, and two-sided conjugacy of higher-rank graphs in terms of the $C^*$-algebras and the Kumjian-Pask algebras of the higher-rank graphs.