RECONSTRUCTING GROUP ACTIONS

Abstract

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" [Formula: see text] from the action we build a group [Formula: see text] of the data and a space [Formula: see text] with an action of [Formula: see text] on [Formula: see text] that arise naturally from the data [Formula: see text]. 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 [Formula: see text] is distinguished by the property that it may be chosen from the action locally. For a large enough set of local data [Formula: see text], our definition of [Formula: see text] 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 [Formula: see text] is chosen from a graph of groups, the group [Formula: see text] is the fundamental group of the graph of groups and the space [Formula: see text] 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.