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.
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ISBN-13
978-3-642-61590-0 (9783642615900)
DOI
10.1007/978-3-642-61590-0
Schweitzer Klassifikation
I. Fractional Linear Transformations.- I.A. Basic Concepts.- I.B. Classification of Fractional Linear Transformations.- I.C. Isometric Circles.- I.D. Commutators.- I.E. Fractional Reflections.- I.F. Exercises.- II. Discontinuous Groups in the Plane.- II.A. Discontinuous Groups.- II.B. Area, Diameter, and Convergence.- II.C. Inequalities for Discrete Groups.- II.D. The Limit Set.- II.E. The Partition of C.- II.F. Riemann Surfaces.- II.G. Fundamental Domains.- II.H. The Ford Region.- II.I. Precisely Invariant Sets.- II.J. Isomorphisms.- II.K. Exercises.- II.L. Notes.- III. Covering Spaces.- III.A. Coverings.- III.B. Regular Coverings.- III.C. Lifting Loops and Regions.- III.D. Lifting Mappings.- III.E. Pairs of Regular Coverings.- III.F. Branched Regular Coverings.- III.G. Exercises.- IV. Groups of Isometries.- IV.A. The Basic Spaces and their Groups.- IV.B. Hyperbolic Geometry.- IV.C. Classification of Elements of Cn.- IV.D. Convex Sets.- IV.E. Discrete Groups of Isometries.- IV.F. Fundamental Polyhedrons.- IV.G. The Dirichlet and Ford Regions.- IV.H. Poincaré's Polyhedron Theorem.- IV.I. Special Cases.- IV.J. Exercises.- IV.K. Notes.- V. The Geometric Basic Groups.- V.A. Basic Signatures.- V.B. Half-Turns.- V.C. The Finite Groups.- V.D. The Euclidean Groups.- V.E. Applications to Non-Elementary Groups.- V.F. Groups with Two Limit Points.- V.G. Fuchsian Groups.- V.H. Isomorphisms.- V.I. Exercises.- V.J. Notes.- VI. Geometrically Finite Groups.- VI. A. The Boundary at Infinity of a Fundamental Polyhedron.- VI.B. Points of Approximation.- VI.C. Action near the Limit Set.- VI.D. Essentially Compact 3-Manifolds.- VI.E. Applications.- VI.F. Exercises.- VI.G. Notes.- VII. Combination Theorems.- VII.A. Combinatorial Group Theory - I.- VII.B. Blocks and Spanning Discs.-VII.C. The First Combination Theorem.- VII.D. Combinatorial Group Theory - II.- VII.E. The Second Combination Theorem.- VII.F. Exercises.- VII.G. Notes.- VIII. A Trip to the Zoo.- VIII.A. The Circle Packing Trick.- VIII.B. Simultaneous Uniformization.- VIII.C. Elliptic Cyclic Constructions.- VIII.D. Fuchsian Groups of the Second Kind.- VIII.E. Loxodromic Cyclic Constructions.- VIII.F. Strings of Beads.- VIII.G. Miscellaneous Examples.- VIII.H. Exercises.- VIII.I. Notes.- IX. B-Groups.- IX.A. An Inequality.- IX.B. Similarities.- IX.C. Rigidity of Triangle Groups.- IX.D. B-Group Basics.- IX.E. An Isomorphism Theorem.- IX.F. Quasifuchsian Groups.- IX.G. Degenerate Groups.- IX.H. Groups with Accidental Parabolic Transformations.- IX.I. Exercises.- IX.J. Notes.- X. Function Groups.- X.A. The Planarity Theorem.- X.B. Panels Defined by Simple Loops.- X.C. Structure Subgroups.- X.D. Signatures.- X.E. Decomposition.- X.F. Existence.- X.G. Similarities and Deformations.- X.H. Schottky Groups.- X.I. Fuchsian Groups Revisited.- X.J. Exercises.- X.K. Notes.- Special Symbols.
