Subject Index

The following index does not contain the terms which we presume are well­known to the reader, even if they appear as definitions in the book (e.g. the 'signature' of a bi-linear form). For the terms appearing most frequently in the book (e.g. 'hyperbolic manifold') we indicate the first occurrence only (i. e. the page where they are defined); these terms are preceded by an *asterisk. In general, the boldface numeral indicates the page where the term is defined, and the normal numeral indicates further occurrences. A few terms are defined more than once; of course the definitions are always consistent: they usually refer to slightly different cases or generalizations.

affine (manifold) 55, 286, 292, 297-298, 303-304, 307

algebraic volume (of a simplex) 107, 203

*amenable group 305 asymptot.ically parallel (subspaces of

lin) 30, 32, 71, 90-92 at.oroidal (manifold) 156

boundary of lIn 29,31,34, 78, 85-88, 92, 125

*Chaballty topology 160 complete (in a non-metric sense) 55 conformal mapping 7-11, 15, 18, 97 conformal structure 7, 20-25, 27, 29,

34 ,36 *cup product 276

curvature 38-43, 51-53, 58-62, 126-127, 133, 139-140, 156, 312

cusp 145, 150-157,159, 174-182, 185, 192-193,197-198,202,251,267,271

*deformations (of a complete hyper­bolic structure) 251

Dehn surgery (on a manifold) 191, 198, 264

Dehn surgery (along a link) 192-193, 271

developing map 47, 52-54, 55, 230, 232, 257-265

*disc model of lIn 2

elliptic (isometry of - type) 31-34, 65, 126, 141, 146, 163-164

elliptic (manifold) 46, 51, 54-55, 58-61, 127, 157

*end (of a hyperbolic manifold) 143 Euclidean (or flat, manifold) 46, 141,

145,148-151,206,229-230,233,252-257, 268, 286

*Euler number (of a flat fiber bundle) 291

exponential growth (of a group) 318

322 Subject Index

*flat fiber bundle 55, 127, 281 flat (manifold) 46, 51, 54-55, 58-61,

127, 156-157, 189, 286, 312 foliation 281-282, 286

*freely (group operating -) 49 free-homotopic (loops) 63 fundamental class 104, 279 fundamental domain 85-86, 110, 113,

116, 152-155, 163, 207, 260, 316

*geometric topology 165, 167 Gromov norm 105, 127, 130-131, 268,

278,292

*half-space model (of Hn) 2 holonomy 54, 127, 164, 230-233, 257,

260 holonomy (of a flat fiber hundle) 284-

286, 295, 197-301, 304-307 homotopy equivalence 83, 119, 123-

124, 302-303, 320 homot.opy inverse 83, 86, horohall 35, 202, 205-206, 311-312 horosphere 34-35, 97, 141, 205,

*hyperbolic Dehn surgery (wpak and strong) 192

hyperbolic (isometry of - t.ype) 31-34, 65, 125-126, 129-130, 141, 145-150,311

*hyperbolic (manifold) 46 *hyperbolic n-space 1 hyperbolic suhspace 26-30, 85-86, 90-

92, 106, 116 *hyperholoid model of Hn 1

ideal (simplex) 94-98, 102, 113, 121, 197,201-204,207,224-225,238,251, 260, 266-267, 270

IEP (isometries-extension propert.y) 52-53

incompressible 156, 269, 271 infinity (points at -) 29-30,31, 40,

66-67, 141, 145, 150 illjectivity radius 166, 174

inversion 2, 7-8, 13-16, 20-29, 35-36, 96, 121, 135

irreducihle 156, 210, 271-272 isotopic (curves) 65-69 isotopic (diffeomorphisms) 62-63, 68,

76, 81, 128, 147, 295

*Klein model of lIn 2

Lobachevsky function 98, 267

modular group 81, 126 *modulns (ohn ideal tetrahedron) 225

order (of a manifold in Fn) 192-l!H

pant, pant decomposition (of a sur­face) 56-80

parabolic (isometry of - type) 31-34, 65, 126, 140-142, 145-150, 157, 202, 205,251

paradoxal partition 317 peripheral 156 polynomial growth (of a group) 317 prime (manifold) 157

*projective model of H n 2 *properly dicontinllollsly (group oper­

ating -) 49 pseudo-isometry 85-87, 91 pSf'udo-sphC're 42

quasi-isometry 168,173.175, Iii, 195

reflection 5-23, 32, 116, 135 regular (simplC'x) 94, 96, 102, 106,

112-115, 121

similarity (structure) 229-233, 252-261,

standard spine 208-210, 272 *(X, G)-st.ructure 45 *straight (simplex) 106 sub-exponential growth (of a group)

318

surgery (.~ee Dehn surgery or hyper­bolic Dehn surgery)

*thick part 133 *thin part 133

topological end 143-144 tractrix 41 tube 143, 150, 153-154, 177-182, 193,

205

Subject Index 323

ultra-parallel (subspaces of Hn ) 30, 32

*weak equivalence (of fiber bundles) 127-128, 281

Notation Index

For t.he readf'r's convemence we list. the non-st.andard notations most. fre­quently used in the book, and explain t.h('ir meaning; we omit t.he symbols whose meaning is self-evid('nt (such as 'Conf(.U)' for the group of conformal automorphisms of a manifold _H).

£(0

A

Boundary of hyperholic 11-space

set. of all discrete torsion-free suhgronps of a t.opological group G

Disc model of hyperholic n-space

Eulf'r class of a flat vector hundk ,

The n-t.h Margnlis constant

Family of all fillite-volnmf' complde ori('nted hyperholie n-manifolds

Family of element.s of Tn haying volume at most c

Family af all n-dimf'nsional complet.e orif'nted hyperholic manifolds

(Not necessarily compld,(') hyperholic strndll1'es snpported hy a manifold M of 73 Hyperholic n-space (as an abstract Ri('mannian manifold)

Hyperholic n-space together with its houndary

Inversion wit.h respect to the sph('re of cent.re 1'0 and radius va Hyperholoid modd of hyperbolic n-space

Group of isometries of a Riemannian manifold Al

Group of orientat.ion-preserying isometries of an orif'nted Rieman­nian manifold !VI

Lohachevsky function

!VId, " .. ,dk Manifold obtained by Dchn Sll1'gery of coefficients (It, .. " (h from a manifold AI whose houndary consists of tori

AI(O,e) The c:-thin part of a hypf'rholic manifold iU

AI[e,co) The c:-thick part of a hyperbolic manifold Al

IIMII Gromm' norm of a manifold M

Px,y Parallel transport along the unique geodesic line joining two points x and y of 1Hn

Notation IlHi('x 325

lIIn ,+ Half-space model of hyperbolic l1-space

Sn Set of geodesic simplices in Hn

Tg Compact oriented surface of genus g

T3 Family of oriented three-manifolds bounded hy tori and ohtained by gllleing tetrahedra along faces and removing vertices

Tg Teichmiiller space (hyperholic structures on a surface of genus g, up to isometries isotopic to the identity)

Vn Maximal volume of a geodesic simpl('x in 1Hn

<I: ( V, 11') Angle between two vectors v and tv

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Kloeden, P.; Ombach, f.; Cyganowski, S.: From Elementary Probability to Stochastic Differential Equations with MAPLE

Kloeden, P. E.; Platen; E.; Schurz, H.: Nu­merical Solution of SDE Through Computer Experiments

Kostrikin, A. 1.: Introduction to Algebra

Krasnoselskii, M. A.; Pokrovskii, A. v.: Sys­tems with Hysteresis

Luecking, D. H., Rubel, L. A.: Complex Anal­ysis. A Functional Analysis Approach

Ma, Zhi-Ming; Roeckner, M.: Introduction to the Theory of (non-symmetric) Dirichlet Forms

Mac Lane, S.; Moerdijk, 1.: Sheaves in Geom­etry and Logic

Marcus, D. A.: Number Fields

Martinez, A.: An Introduction to Semiclas­sical and Microlocal Analysis

Matousek, J.: Using the Borsuk-Ulam Theo­rem

Matsuki, K.: Introduction to the Mori Pro­gram

Mc Carthy, P. J.: Introduction to Arithmeti­cal Functions

Meyer, R. M.: Essential Mathematics for Ap­plied Field

Meyer-Nieberg, P.: Banach Lattices

Mines, R.; Richman, F.; Ruitenburg, w.: A Course in Constructive Algebra

Moise, E. E.: Introductory Problem Courses in Analysis and Topology

Montesinos-Amilibia, J. M.: Classical Tessel­lations and Three Manifolds

Morris, P.: Introduction to Game Theory

Nikulin, V. v.; Shafarevich, I. R.: Geometries and Groups

aden, J. J.; Reddy, J. N.: Variational Methods in Theoretical Mechanics

0ksendal, B.: Stochastic Differential Equa­tions

Poizat, B.: A Course in Model Theory

Polster, B.: A Geometrical Picture Book

Porter, J. R.; Woods, R. G.: Extensions and Absolutes of Hausdorff Spaces

Radjavi, H.; Rosenthal, P.: Simultaneous Tri­angularization

Ramsay, A.; Richtmeyer, R. D.: Introduction to Hyperbolic Geometry

Rees, E. G.: Notes on Geometry

Reisel, R. B.: Elementary Theory of Metric Spaces

Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods

Ribenboim, P.: Classical Theory of Algebraic Numbers

Rickart, C. E.: Natural Function Algebras

Rotman, J. J.: Galois Theory

Rubel, L. A.: Entire and Meromorphic Func­tions

Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations

Sagan, H.: Space-Filling Curves

Samelson, H.: Notes on Lie Algebras

Schiff, J. L.: Normal Families

Sengupta, J. K.: Optimal Decisions under Uncertainty

Seroul, R.: Programming for Mathemati­cians

Seydel, R.: Tools for Computational Finance

Shafarevich, 1. R.: Discourses on Algebra

Shapiro, J. H.: Composition Operators and Classical Function Theory

Simon net, M.: Measures and Probabilities

Smith, K. E.; KahanpiUi, L.; Kekiiliiinen, P.; Traves, w.: An Invitation to Algebraic Ge­ometry

Smith, K. T.: Power Series from a Computa­tional Point of View

Smorynski, c.: Logical Number Theory I. An Introduction

Stichtenoth, H.: Algebraic Function Fields and Codes

Stillwell, J.: Geometry of Surfaces

Stroock, D. w.: An Introduction to the The­ory of Large Deviations

Sunder, V. S.: An Invitation to von Neumann Algebras

Tamme, G.: Introduction to Etale Cohomol­ogy

Tondeur, P.: Foliations on Riemannian Man­ifolds

Verhulst, F.: Nonlinear Differential Equa­tions and Dynamical Systems

Wong, M. w.: Weyl Transforms

Xamb6-Descamps, S.: Block Error-Cor­recting Codes

Zaanen, A.C.: Continuity, Integration and Fourier Theory

Zhang, F.: Matrix Theory

Zong, C.: Sphere Packings

Zong, C.: Strange phenomena in Convex and Discrete Geometry