Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics)

Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics)

John G. Ratcliffe

Language: English

Pages: 768

ISBN: 038794348X

Format: PDF / Kindle (mobi) / ePub


This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180 illustrations.

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5 Isometries of Hyperbolic Space §5.1. Topological Groups §5.2. Groups of Isometries .. . §5.3. Discrete Groups . . . . . . §5.4. Discrete Euclidean Groups §5.5. Elementary Groups §5.6. Historical Notes . . . . . . 148 148 154 161 169 180 189 6 Geometry of Discrete Groups §6.1. The Projective Disk Model. §6.2. Convex Sets . . . . §6.3. Convex Polyhedra . . . §6.4. Polytopes . . . . . . . §6.5. Fundamental Domains §6.6. Convex Fundamental Polyhedra §6.7. Tessellations .. §6.8. Historical Notes

then 111111 - h(r, Q)I < 1. 3. Hyperbolic Geometry 78 Hence, for all Q :s; P, we have Thus, is rectifiable. By Taylor's theorem, we have T)2 cosh T) :s; 1 + 2 T)4 + 24 cosh T). Hence, if cosh T)(x, y) :s; 12, we have Ilx - yll :s; T)(x, yhh Now suppose that, is rectifiable and [a, b] such that E + T)2(X, y). > O. Then there is a partition P of Let 8 > 0 and set /-lb,8) =supbb(s),,(t)): Is-tl:s; 8}. As , is uniformly continuous, /-lb,8) goes to zero with 8. Hence, there is a 8 > 0 such

line of H2 containing x and y, and let H(x, y, z) be the closed half-plane of H2 with L(x, y) as its boundary and z in its interior. The hyperbolic trtangle with vertices x, y, z is defined to be T(x, y, z) = H(x, y, z) n H(y, z, x) n H(z, x, y). We shall assume that the vertices of T(x, y, z) are labeled in negative order as in Figure 3.5.1. 84 3. Hyperbolic Geometry y z x Figure 3.5.1. A hyperbolic triangle T(x, y, z) Let [x, y] be the segment of L(x, y) joining x to y. The sides of

can identify the group I(Bn) of isometries of the conformal ball model with the group M(Bn) of Mobius transformations of Bn. In particular, we have the following corollary. Corollary 1. The groups I(Bn) and M(Bn) are zsomorphzc. An m-sphere of En is defined to be the intersection of a sphere S(a, r) of En with an (m+ I)-plane of En that contains the center a. An rr::-sph!;-re of En is defined to be either an m-sphere or an extended m-plane P of En. Lemma 2. The group M(En) acts transitively on

of M(Bn). (1) The group G is said to be of ellzptic type if and only if G has a finite orbit in Bn. (2) The group G is said to be of parabolic type if and only if G fixes a point of sn-l and has no other finite orbits in Bn. (3) The group G is said to be of hyperbolic type if and only if G is neither of elliptic type nor of parabolic type. Let cp be in M( Bn) and let x be a point of Bn. Then (cpGcp-l)cp(X) = cp(Gx). In other words, the cpGcp-l-orbit through cp(x) is the 4>-image ofthe G-orbit

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