Infinite group actions on spheres (Q908554)

This survey article centers around questions of the following type: what topological or geometrical properties characterize the infinite conformal or Möbius groups acting on the n-sphere \(S^ n\); what properties determine the structure of their limit sets; are the classical (decomposition/combination, finiteness) theorems about Kleinian groups analytical or topological/geometrical in nature? These questions are discussed in terms of the 3 classes of groups: conformal, quasiconformal, convergence groups; here a group of homeomorphisms of \(S^ n\) is called a convergence group if every sequence of elements has a subsequence converging against a homeomorphism or a constant map (the ``normal family property'' of conformal and quasiconformal homeomorphisms). The main question then is: under what conditions is a quasiconformal or convergence group conjugate to a conformal group? The paper (reserving a chapter for each of the 3 cases \(S^ 1\), \(S^ 2\) and higher dimensions and giving sketches of proofs of some of the main known results) contains a wealth of information and related problems about 3-manifolds, 4- dimensional surgery, negatively curved manifolds and Teichmüller theory, too numerous to be stated here in detail.