Read e-book online Kleinian Groups (Grundlehren der mathematischen PDF

By Bernard Maskit

the fashionable thought of Kleinian teams begins with the paintings of Lars Ahlfors and Lipman Bers; particularly with Ahlfors' finiteness theorem, and Bers' statement that their joint paintings at the Beltrami equation has deep implications for the idea of Kleinian teams and their deformations. From the viewpoint of uniformizations of Riemann surfaces, Bers' commentary has the end result that the query of figuring out the several uniformizations of a finite Riemann floor poses a merely topological challenge; it truly is autonomous of the conformal constitution at the floor. The final chapters the following provide a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We rigorously skirt Ahlfors' finiteness theorem. For teams which uniformize a finite Riemann floor; that's, teams with an invariant part, one could both commence with the idea that the gang is finitely generated, after which use the finiteness theorem to finish that the gang represents basically finitely many finite Riemann surfaces, or, as we do the following, one could begin with the idea that, within the invariant part, the gang represents a finite Riemann floor, after which, utilizing primarily topological strategies, achieve an identical end. extra lately, invoice Thurston wrought a revolution within the box through exhibiting that you possibly can examine Kleinian teams utilizing third-dimensional hyperbolic geome­ attempt, and there's now an lively college of study utilizing those methods.

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Kleinian Groups (Grundlehren der mathematischen Wissenschaften) by Bernard Maskit

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