Geometric finiteness and rigidity of Kleinian groups
Lars V. Ahlfors' work and ideas on the Kleinian groups play an important part in the latest related topics. In this paper the author follows Ahlfors' work and describes the recent developments in the Kleinian groups. NEWLINENEWLINENEWLINEOne of the topics mainly dealt with is the Ahlfors finiteness theorem as follows: let \(G\) be a finitely generated non-elementary Kleinian group. Then, for the ordinary set \(\Omega(G)\) of \(G\), the quotient \(\Omega(G)/G\) is a finite union of conformally finite hyperbolic Riemann surfaces with signature, that is, each Riemann surface may have finitely many punctures or branch points. The author divides the proof into three parts: (a) Each Riemann surface has finite topological type and finitely many surfaces are not conformally equivalent to the thrice-punctured sphere, (b) there are only finitely many thrice punctured spheres, (c) each border component is a puncture. For each part the author gives several proofs and more results. For example, Greenberg's way that \(G\) has a normal subgroup of finite index with no elements of trace \(\neq 2\), Bers' way in relation to the dimension of the space of harmonic Beltrami coefficients, Bers' area theorem, and Sullivan's way in relation to prime ends. NEWLINENEWLINENEWLINEAnother topic, which is still open, is the Ahlfors measure problem whether the limit set of a finitely generated Kleinian group has a zero area. The author describes Ahlfors' original proof for the geometrically finite case and refers to Thurston's approach, using the visual average of the characteristic function of the limit set.NEWLINENEWLINEFor the entire collection see [Zbl 0944.00101].
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