Orbit equivalence of graphs and isomorphism of graph groupoids

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.

9 pages. In version 3 the list of references has been updated and some misprints have been corrected. This is the version that will be published

References in corpus (5)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras

Cited by in corpus (8)

• Reconstruction of groupoids and C*-rigidity of dynamical systems
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• C*-algebras, groupoids and covers of shift spaces
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• Orbit equivalence of higher-rank graphs
• $C^{\ast }$-rigidity of dynamical systems and étale groupoids