Groupoid actions on topological spaces and Bass-Serre theory

Abstract: The so-called Bass-Serre theory gives a complete and satisfactory description of groups acting on trees via the structure theorem. We construct a Bass-Serre theory in the groupoid setting and prove a structure theorem. Groupoids are algebraic objects that behave like a group (i.e., they satisfy conditions of associativity, left and right identities and inverses) except that the multiplication operation is only partially defined. Any groupoid action without inversion of edges on a forest induces a graph of groupoids, while any graph of groupoids satisfying certain hypothesis has a canonical associated groupoid, called the fundamental groupoid, and a forest, called the Bass–Serre forest, such that the fundamental groupoid acts on the Bass–Serre forest. The structure theorem says that these processes are mutually inverse, so that graphs of groupoids "encode" groupoid actions on forests. One of the main differences between the classical setting and the groupoid one is the following: in the classical setting, given a group action without inversion on a graph, one of the ingredients used to build a graph of groups is the quotient graph given by such action; in the groupoid context, there is not a canonical graph associated to the action of a groupoid on a graph. Hence, we need to resort to the difficult notion of desingularization of a groupoid action on a graph.

group theoryrings and algebras

Audience: researchers in the topic

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