Finitely generated non-cocompact NEC groups

Abstract

This thesis is devoted to the study of finitely generated discrete subgroups r of the whole group of isometries of the hyperbolic plane H including those which reverse the orientation (reflections and glide reflections) as well as boundary transformations (parabolic and boundary hyperbolic elements), such that the orbit space H/r is not compact. Two specia l cases closely related to finitely generated non-cocompact NEC groups, the finitely generated discrete subgroups of orientation-preserving isomet ries (fuchsian groups) and the cocompact NEC groups have been extensively studied in the literature. This work presents a fairly complete introduction of the non-cocompact NEC groups, providing with proof their presentation, int roducing their signa tures and using them for st udying their orbit spaces and the necessary and sufficient conditions of isomorphism between these groups. We present additionally a set of inva ria nts that classify the non-compact Klein surfaces up to homeomorphisms using the signature of the NEC group of which the Klein surface is the orbit space. The Euler characteristic of the orbit space of an NEC group is calculated. Using this we obtain the signa ture of the non-cocompact canonical fuchsian group linked to the sign at ure of a given NEC group. Finally, the concept of elementary NEC groups is int roduced and all the possible elementary groups deduced. Using the properties of their canonical fuchsian groups, sorne result s describing the limit sets of NEC groups are obtained. That leads us to int roduce a classification of NEC groups of first and second kind similarly as for fuchsian groups. Keywords: Hyperbolic Plane, Non-euclidean Chryst allogra phic Gro ups, Finitely generated Groups of Hyperbo lic Isometries, Non -cocompact NEC Groups