Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras

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Given a graph E, an action of a group G on E, and a G-valued cocycle ϕ on the edges of E, we define a C*-algebra denoted OG,ₑ, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup SG,ₑ built naturally from the triple (G,E,ϕ). As a tight C*-algebra, OG,ₑ is also isomorphic to the full C*-algebra of a naturally occurring groupoid Gtight(SG,ₑ). We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which OG,ₑ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our OG,ₑ, and many of their known properties are shown to follow from our general theory.