Orbit equivalence of graphs and isomorphism of graph groupoids

TL;DR

This paper establishes a precise correspondence between the isomorphism of graph groupoids and orbit equivalence of directed graphs, including conditions involving isolated points and graph stabilizations.

Contribution

It provides a complete characterization of when graph groupoids are isomorphic based on orbit equivalence, especially for graphs with finitely many vertices and no sinks.

Findings

Groupoids of two directed graphs are isomorphic iff the graphs are orbit equivalent with preserved isolated periodic points.

Isomorphism of groupoids for graphs with finitely many vertices and no sinks is equivalent to their orbit equivalence.

Stabilizations of such graphs have isomorphic groupoids iff the stabilized graphs are orbit equivalent.

Abstract

We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the (topological) isolated points of the boundary path space of a graph. As a result, we are able to show that the groupoids of two directed graphs with finitely many vertices and no sinks are isomorphic if and only if the two graphs are orbit equivalent, and that the groupoids of the stabilisations of two such graphs are isomorphic if and only if the stabilisations of the graphs are orbit equivalent.