On the Algebraic Structure of Discontinuous Groups

The purpose of this paper is to determine what may be said, under favorable conditions, about the algebraic structure of a group of homeomorphisms when a fundamental domain is known. If S is a space and G a group of homeomorphisms acting on S, then the image of a point x in 5 under an element g in G will be denoted by gx. If D is a subset of S, then gD will be the image of D, and if F is a subset of G, then Fx will denote the union of the images gx of x and FD will denote the union of the sets gD for all g in F. The empty set will be denoted by 0, the identity element of a group by 1, and intersections, inclusions, and closures will be indicated in the usual ways. Let there be given a connected and locally connected Hausdorff space S, a group G of homeomorphisms acting on S, and a connected subspace D oí S satisfying the following conditions: 1. 5 is the disjoint union of the sets gD with gEG. 2. Let F= {gEG\gDC\D?¿0\. Then the number of elements in F is finite and FD contains a neighborhood of D. If the foregoing conditions are satisfied, then in order that these concepts have names, S will be called an admissible space, G a group of the first kind acting on S, and D a proper fundamental domain for G. It can be verified without great difficulty that the elements of F generate G,2 and F will be called the local set of generators relative to D. D will be fixed throughout, and the phrase "relative to D" will generally be omitted. It is evident that 1£F and that gEF implies g~lEF. If the space 5 is simply connected, then the relations which hold between the local generators can be found. To state the main theorem.precisely it is necessary to introduce certain groups of which G is a homomorphic image. Let Ë be an abstract free group the generators of which are the elements of F different from the identity. To avoid confusion it will be assumed that a mapping o* is given which takes the elements of F different from the identity, considered as a subset of G, onto the generators of Ë, and that o is extended to all of F by setting <r(l) = 1. In Ë form the smallest normal subgroup R containing all the elements