Math Ann 237 253—272 (1978) © by Springer Verlag 1978

Groups with Domains of Discontinuity

R S Kulkarni*

Department of Mathematics, Indiana University, Bloomington, IN 47401, USA

The classical theory of Kleinian groups deals with discrete subgroups of the group of Mobius transformations acting properly discontinuously on some nonempty open subset of the Riemann sphere The special and better known case is the theory of Fuchsian groups where the domain is the interior of a disc which may be identified with the upper hemisphere In the last fifteen years the interest in the general case has been revived thanks to the work of Ahlfors, Bers, Maskit, and others An outstanding problem in the theory of Kleinian groups is to find its higher dimensional extensions, e g in the study of discrete subgroups of holomor- phic transformations acting properly discontinuously on a nonempty domain in a compact hermitian symmetric space For a survey until 1952 see Borel [6], especially §V A major part of these researches are extensions of the Fuchsian case However, the general Kleiman case has also been studied, cf Myrberg [17-19], Schubart [20] , for a conformai geometric extension see Ahlfors [2, 3], etc These researches are mainly analytic in nature and a major difficulty which has been clearly recognized is the lack of strong uniformization theorems in higher dimensions

On the other hand some of the major results of the classical theory, especially Ahlfors' finiteness theorem [1], Bers' area theorems [4], Klein's and Maskit's combination theorems [14], etc have a striking topological flavor which, one hopes, should have some higher dimensional generalizations As it turns out the theory of ends of very general topological spaces due to Freudenthal [9] and the theory of ends of abstract, finitely generated groups as developed by Hopf, Freudenthal, Stalhngs, etc (cf [12, 10, 21]) suffice to explain the topological genesis of the classical theory and also provide a first step towards its higher dimensional generalizations Broadly speaking, the action of a properly tinuous group at the ends of a space contains some gross "C^-information" which one should attempt to seek first before asking for finer "analytic information" This paper is an attempt to make this feeling more concrete and precise

* Partially supported by NSF-GPS 71-03442

0025 - 5831 / 78 / 0237 / 0253 / $04 00