A groupoid -algebraic Bass Serre theorem.

The concept of a group action on a space was generalized to a groupoid action and it has applications to dynamical systems, representation theory and operator algebras. If groups can roughly be described as the set of symmetries of certain objects, then groupoids can be thought as the set of symmetries of fibered objects. Thanks to the knowledge of Bass-Serre theory for groups, we were able to establish a Bass-Serre theory for groupoids. A graph of groupoids is given by a connected graph together with a groupoid for each vertex and edge of , and monomorphisms from each edge groupoid to the adjacent vertex groupoid.

To prove a groupoid -algebraic Bass-Serre theorem we associate to a graph of groupoids a groupoid, called the universal fundamental groupoid, and a forest, called the universal forest, on which the universal fundamental groupoid acts. Such an action induces an action of the universal fundamental groupoid on the boundary of the universal forest. Given a locally-finite nonsingular graph of groupoids, we work with two -algebras: the graph of groupoids -algebra, which is universal for generators and relations encoding an underlying combinatorial object, and the action groupoid -algebra induced by the action of the universal fundamental groupoid on the boundary of the universal forest. We prove that the two algebras are isomorphic.