Reconstructing group actions

We give a general structure theory for reconstructing non-trivial group actions on sets without any further assumptions on the group, the action, or the set on which the group acts. Using certain "local data" D from the action we build a group G(D) of the data and a space X(D) with an action of G(D) on X(D) that arise naturally from the data D. We use these to obtain an approximation to the original group G, the original space X and the original action G × X → X. The data D is distinguished by the property that it may be chosen from the action locally. For a large enough set of local data D, our definition of G(D) in terms of generators and relations allows us to obtain a presentation for the group G. We demonstrate this on several examples. When the local data D is chosen from a graph of groups, the group G(D) is the fundamental group of the graph of groups and the space X(D) is the universal covering tree of groups. For general non-properly discontinuous group actions our local data allows us to imitate a fundamental domain, quotient space and universal covering for the quotient. We exhibit this on a non-properly discontinuous free action on . For a certain class of non-properly discontinuous group actions on the upper half-plane, we use our local data to build a space on which the group acts discretely and cocompactly. Our combinatorial approach to reconstructing abstract group actions on sets is a generalization of the Bass-Serre theory for reconstructing group actions on trees. Our results also provide a generalization of the notion of developable complexes of groups by Haefliger.