Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic KnotsAmerican Mathematical Soc., 14.07.2009 - 384 Seiten The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified. This book is published in cooperation with IAS/Park City Mathematics Institute. |
Inhalt
| 1 | |
The hyperbolic plane | 11 |
The 2dimensional sphere | 47 |
Gluing constructions | 55 |
Gluing examples | 89 |
Tessellations | 133 |
Group actions and fundamental domains | 185 |
The Farey tessellation and circle packing | 207 |
The 3dimensional hyperbolic space | 227 |
Kleinian groups | 241 |
The figureeight knot complement | 293 |
Geometrization theorems in dimension 3 | 315 |
Appendix Tool Kit | 355 |
Supplemental bibliography and references | 365 |
| 377 | |
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Low-dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots Francis Bonahon Keine Leseprobe verfügbar |
Häufige Begriffe und Wortgruppen
angle antilinear antilinear fractional map ball Bdhyp Bd(P centered complete geodesic consequence consider contained converges corresponding crooked Farey tessellation defined definition delimited discrete walk disjoint disk sector edge element endpoints euclidean metric euclidean plane Exercise exists Figure finite finitely Ford domains fundamental domain geodesic geodesic g geometric gluing maps H2,d hyp homeomorphism homotheties horizontal translation horocircle horosphere hyperbolic isometry hyperbolic length hyperbolic metric hyperbolic plane hyperbolic plane H2 hyperbolic space hyperbolic space H3 ideal vertex infimum inversion isometry isometry of H2 Klein bottle kleinian group knot limit set line segment linear fractional map metric space namely parametrized piecewise differentiable curve plane H2,d Pn+1 point P0 polygon proof of Theorem Proposition prove quotient metric quotient space radius Riemann sphere rotation Section sends sequence Show sphere spherical subset tangent tiling group torus Triangle Inequality vector vertices ΛΓ