Book contents
- Frontmatter
- Contents
- Preface
- Symmetries of modular surfaces
- Lifting group actions to covering spaces
- A combinatorial approach to the symmetries of M and M–l Riemann surfaces
- Inequalities for Pell equations and Fuchsian groups
- The Euler characteristic of graph products and of Coxeter groups
- Infinite families of automorphism groups of Riemann surfaces
- Planar hyperelliptic Klein surfaces and fundamental regions of NEC groups
- An example of an infinite group
- Moduli of Riemann surfaces with symmetry
- Modular groups – geometry and physics
- On automorphisms of free products
- The growth series of the Gieseking group
- Exceptional representations of PSL2(q) of monodromy genus zero
- On the rank of NEC groups
- The geometry of bending quasi-Fuchsian groups
- Farey series and sums of continued fractions
- Commensurability classes of two-generator Fuchsian groups
- Limit points via Schottky pairings
- Diagonalizing Eisenstein series III
- Some remarks on 2-generator hyperbolic 3-manifolds
- Uniformization, graded Riemann surfaces and supersymmetry
- Generating sets for finite groups
- Group actions on trees with and without fixed points
Modular groups – geometry and physics
Published online by Cambridge University Press: 10 December 2009
- Frontmatter
- Contents
- Preface
- Symmetries of modular surfaces
- Lifting group actions to covering spaces
- A combinatorial approach to the symmetries of M and M–l Riemann surfaces
- Inequalities for Pell equations and Fuchsian groups
- The Euler characteristic of graph products and of Coxeter groups
- Infinite families of automorphism groups of Riemann surfaces
- Planar hyperelliptic Klein surfaces and fundamental regions of NEC groups
- An example of an infinite group
- Moduli of Riemann surfaces with symmetry
- Modular groups – geometry and physics
- On automorphisms of free products
- The growth series of the Gieseking group
- Exceptional representations of PSL2(q) of monodromy genus zero
- On the rank of NEC groups
- The geometry of bending quasi-Fuchsian groups
- Farey series and sums of continued fractions
- Commensurability classes of two-generator Fuchsian groups
- Limit points via Schottky pairings
- Diagonalizing Eisenstein series III
- Some remarks on 2-generator hyperbolic 3-manifolds
- Uniformization, graded Riemann surfaces and supersymmetry
- Generating sets for finite groups
- Group actions on trees with and without fixed points
Summary
Introduction
Recent developments in the study of discrete groups associated to Riemann surfaces serve to emphasize once again the importance of geometric ideas in understanding structural properties of the most abstract kind. We consider here two instances of this maxim applied to the mapping-class groups Modg,n,i each occurs within the action as the modular group on the corresponding Teichmuller space, Tgn, which classifies marked complex structures on an n-pointed genus g surface. This space carries intrinsically a structure of complex manifold with dimension 3g + n – 3 and a complete global metric d defined in terms of the least (logarithmic) distortion necessary in deforming one complex structure to another. The modular group action is isometric and serves to identify points of Tg,n that represent holomorphically equivalent structures.
The first question we discuss concerns geometric structures of a more concrete type within this metric framework, defined by a particular kind of deformation which one now calls a Teichmüller geodesic disc, a natural class of complex submanifold of Tg,n isometric to the Poincaré metric model of the hyperbolic plane; they exist in profusion (through an arbitrary point in any given direction). We present a construction via hyperbolic geometry of some examples of Teichmüller discs which project to finite volume Riemann surfaces immersed in Pg; on suitable finite coverings they form smooth totally geodesic submanifolds, isomorphic to affine plane models of certain familiar algebraic curves.
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- Discrete Groups and Geometry , pp. 94 - 103Publisher: Cambridge University PressPrint publication year: 1992
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