Generalizations of Fuchsian and Kleinian Groups

Abstract

We will here consider two classes of groups of homeomorphisms of the com-pactified n-dimensional euclidean space R n = R n∪{∞}, somehow generalizing various aspects of the behaviour of the classical Fuchsian and Kleinian groups whose elements are conformai homeomorphisms of R2. The first of the generalizations is the class of quasiconformal groups relaxing the con-formality condition for elements of Fuchsian and Kleinian groups. The second class of groups, convergence groups, generalizes topological convergence properties of the classical conformai groups. Especially this latter class of groups has some interesting applications as it can be applied to obtain proofs of the Nielsen realization problem and of the Seifert conjecture to be discussed in the final section.