Pekka Tukia: Generalizations of Fuchsian and Kleinian groups

A Kleinian or Fuchsian group is a discrete group of Möbius transformations of the n-sphere. That is, it is a group of conformal homeomorphisms. We will discuss quasiconformal and convergence groups as generalizations of these groups.

A quasiconformal group is a group of quasiconformal homeomorphisms of the n-sphere where one relaxes the assumption on conformality but in a bounded way. Thus the definition still involves some analyticity. In contrast, the definition of a convergence group is purely topological, with starting point some convergence properties of Kleinian groups.

These groups have had some interesting applications.