Reconstructing Group Actions

We give a general structure theory for reconstructing group actions on sets without any 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 R. 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.