Groupoid models for diagrams of groupoid correspondences and their C*-algebras

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Diagrams of groupoid correspondences describe a very general kind of dynamical systems, which contains self-similarities of groups and topological graphs as special cases. Self-Similar groups are important invariants of dynamical systems because they are purely algebraic and still may determine all of the dynamics on a Julia set. Graph C*-algebras are an important class of C*-algebras because many of their properties may be read off the underlying graphs. In order to handle more C*-algebras in similar ways, several generalisations of graph C*-algebras have been introduced in recent years. Diagrams of groupoid correspondences allow to treat all these classes of examples in a uniform way and also provide more examples that are similarly amenable to study, that is, they inherit many important properties of graph C*-algebras. Just as a group action may be encoded in a groupoid, which then gives rise to a C*-algebra, so a diagram of groupoid correspondences may be described in one groupoid, called its groupoid model. The existing constructions of this groupoid, however, only work under an extra assumption. A suitable generalisation to all diagrams is an important goal of this project. Alternatively, one may first translate a diagram of groupoid correspondences to the realm of C*-algebras or plain algebras and then map the diagram to a single C*-algebra or algebra, called its a covariance algebra. How is this covariance algebra related to the groupoid C*-algebra or Steinberg algebra of the groupoid model? This is the second central question in this proposal. It is known that these objects agree under some assumptions, but not always. Finally, we will investigate some important properties of groupoid models, which allow to decide, for instance, whether the groupoid model is simple, that is, its only quotients are the obvious ones.