Kleinian Groups

1. Front Matter

   Pages I-XIII

   PDF

2. Fractional Linear Transformations

   • Bernard Maskit

   Pages 1-14

3. Discontinuous Groups in the Plane

   • Bernard Maskit

   Pages 15-40

4. Covering Spaces

   • Bernard Maskit

   Pages 41-52

5. Groups of Isometries

   • Bernard Maskit

   Pages 53-83

6. The Geometric Basic Groups

   • Bernard Maskit

   Pages 84-114

7. Geometrically Finite Groups

   • Bernard Maskit

   Pages 115-134

8. Combination Theorems

   • Bernard Maskit

   Pages 135-170

9. A Trip to the Zoo

   • Bernard Maskit

   Pages 171-213

10. B-Groups

    • Bernard Maskit

    Pages 214-248

11. Function Groups

    • Bernard Maskit

    Pages 249-318

12. Back Matter

    Pages 319-328

    PDF

The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome­ try, and there is now an active school of research using these methods.

• Area
• Dimension
• Finite
• Group theory
• Invariant
• Riemann surface
• approximation
• convergence
• field
• finite group
• function
• hyperbolic geometry
• mapping
• theorem
• uniformization

• Dept. of Mathematics, SUNY at Stony Brook, Stony Brook, USA

  Bernard Maskit

Policies and ethics