Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras

Abstract

We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu--Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid ${\mathrm{C}}^{\ast }$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu--Renault systems are constructed, each modelling the Cuntz algebra ${\mathcal{O}}_2$. The ${\mathrm{C}}^{\ast }$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into ${\mathcal{O}}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into ${\mathcal{O}}_2\otimes \mathcal{K}$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a ${}^{\ast }$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit ${}^{\ast }$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.

39 pages, one figure