Finitely generated Kleinian groups. An introduction. (English) Zbl 0672.30034

Bers, Lipman

Ann. Acad. Sci. Fenn., Ser. A I, Math. 13, No. 3, 313-327 (1989).

This paper is an introduction to and survey of the theory of finitely generated Kleinian groups. The emphasis is on those parts of the theory which have connections with Riemann surfaces and quasiconformal mapping. After presenting some definitions and examples, there is a discussion of Klein’s combination theorem and the first and second combination theorems of Maskit as well as Maskit’s theorem that a finitely generated Kleinian group is Schottky if and only if it is free and has no parabolic elements.
There is a presentation of Ahlfors’ finiteness theorem which asserts that for a finitely generated Kleinian group the factorization of the ordinary set by the group yields a finite number of Riemann surfaces, each of which is of finite type. Also, there is a discussion of Bers’ area inequalities, which are quantitative refinements of the finiteness theorem. In particular, the first area inequality gives a bound for the total non-Euclidean area of all of the covered Riemann surfaces in terms of the number of generators of the Kleinian group. The second area inequality says that if the Kleinian group G possesses one invariant component \(\Delta\), then the total area of all of the finite number of Riemann surfaces covered by G is less than or equal to twice the area of the Riemann surface covered by \(\Delta\). There are further sections on questions concerning the measure of the limit set, the technique of quasiconformal mapping, the Beltrami equation, Teichmüller spaces, and on Kleinian groups which appear on the boundary of finite dimensional Teichmüller spaces.

Reviewer: F.P.Gardiner

MSC:

30F40

Kleinian groups (aspects of compact Riemann surfaces and uniformization)

Keywords:

finiteness theorem; Schottky groups; quasiconformal mapping; Beltrami equation; Teichmüller spaces