On the algebraic structure of discontinuous groups

Abstract

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 S under an element g in G will be denoted by gx. If 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 of S satisfying the following conditions: 1. S is the disjoint union of the sets gD with gCG. 2. Let F= {gEGIgDnDg 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 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 to D. will be fixed throughout, and the phrase relative to D will generally be omitted. It is evident that 1 C F and that g C F implies g9ECF. If the space S 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 1 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 a is given which takes the elements of F different from the identity, considered as a subset of G, onto the generators of ii, and that a is extended to all of F by setting a(1) = 1. In Ff form the smallest normal subgroup R containing all the elements

Locations

• Proceedings of the American Mathematical Society - View - PDF