Obtaining presentations from group actions without making choices

Consider a group G acting nicely on a simply-connected simplicial complex X . Numerous classical methods exist for using this group action to produce a presentation for G. For the case that X/G is 2-connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.