Geometry, topology and dynamics of character varieties / editors, William Goldman, Caroline Series, Ser Peow Tan.

Format:

Book

Contributor:

Goldman, William.

Series, Caroline.

Tan, Ser Peow.

Series:

Lecture notes series (National University of Singapore. Institute for Mathematical Sciences) ; v. 23.

Lecture notes series. Institute for Mathematical Sciences, National University of Singapore ; v. 23

Language:

English

Subjects (All):

Topology.

Physical Description:

1 online resource (362 p.)

Edition:

1st ed.

Place of Publication:

Hackensack, N.J. : World Scientific, 2012.

Language Note:

English

Summary:

This volume is based on lectures given at the highly successful three-week Summer School on Geometry, Topology and Dynamics of Character Varieties held at the National University of Singapore's Institute for Mathematical Sciences in July 2010. Aimed at graduate students in the early stages of research, the edited and refereed articles comprise an excellent introduction to the subject of the program, much of which is otherwise available only in specialized texts. Topics include hyperbolic structures on surfaces and their degenerations, applications of ping-pong lemmas in various contexts, intro

Contents:

CONTENTS; Foreword; Preface; An Invitation to Elementary Hyperbolic Geometry Ying Zhang; Introduction; 1. Euclid's Elements, Book I and Neutral Plane Geometry; 1.1. A brief review of contents of Elements, Book I; 1.2. A useful lemma; 1.3. A gure-free proof of Proposition I.7; 1.4. More notes on Elements, Book I; 1.5. Playfair's axiom; 1.6. Neutral plane geometry; 1.7. Angle-sums of triangles and Legendre's Theorems; 1.8. Quadrilaterals with two consecutive right angles; 1.9. Saccheri and Lambert quadrilaterals; 1.10. Variation of triangles in a neutral plane

1.11. A midline configuration for triangles1.12. More theorems of neutral plane geometry; 1.13. Small angles; 2. Hyperbolic Plane Geometry; 2.1. Hyperbolic plane; 2.2. Asymptotic Parallelism; 2.3. Angle of parallelism; 2.4. The variation in the distance between two straight lines; 2.5. Some more theorems in hyperbolic plane geometry; 2.6. Construction of the common perpendicular to two ultra-parallel straight lines; 2.7. Construction of asymptotic parallels; 2.8. Ideal points; 2.9. Horocycles; 2.10. Construction of the straight line joining two given ideal points; 2.11. Ultra-ideal points

2.12. The projective plane associated to a hyperbolic plane2.13. Center-pencils of a hyperbolic triangle; 2.14. Equidistant curves; 2.15. Positions of proper points relative to an ideal point; 2.16. Hyperbolic areas via equivalence of triangles; 2.17. Metric relations of corresponding arcs in concentric horocycles; 3. Isometries of the Hyperbolic Plane; 3.1. Isometries and reections in straight lines; 3.2. Orientation preserving/reversing isometries; 3.3. Rotations; 3.4. Translations; 3.5. Isometries of parabolic type; 3.6. Redundancy of two reflections

3.7. Orientation reversing isometries as reflections and glide reflections3.8. Isometries as projective transformations; 3.9. Invariant projective lines of; 3.10. Composition of two orientation preserving isometries other than two translations; 3.11. Composition of two translations; 3.12. Conjugates of isometries; 3.13. The orthic triangle; 4. Hyperbolic Trigonometry Derived from Isometries; 4.1. Some identities of isometries of a neutral plane; 4.2. Some trigonometric formulas in H2(k); 4.3. Upper half-plane model U2 for hyperbolic plane H2(1); 4.4. Matrices of certain isometries of U2

4.5. Trigonometric laws via identities of isometries4.6. Suggested further readings; Acknowledgments; References; Hyperbolic Structures on Surfaces Javier Aramayona; 1. Introduction; 2. Plane Hyperbolic Geometry; 2.1. Mobius transformations; 2.1.1. Classification in terms of trace and fixed points; 2.2. Models for hyperbolic geometry; 2.2.1. Hyperbolic distance; 2.2.2. Mobius transformations act by isometries; 2.2.3. The Cayley transformation; 2.2.4. Hyperbolic geodesics; 2.2.5. The boundary at infinity; 2.2.6. The full isometry group; 2.2.7. Dynamics of elements of Isom+(H)

2.3. Fuchsian groups and fundamental domains

Notes:

Description based upon print version of record.

Includes bibliographical references.

ISBN:

9786613784360

9781281603678

1281603678

9789814401364

9814401366

OCLC:

804661889