index [assets.cambridge.org]assets.cambridge.org/97811071/16740/index/9781107116740_index… ·...
Post on 30-Apr-2020
19 Views
Preview:
TRANSCRIPT
Index
2π Lemma, 359H2,H3; hyperbolic 2D, 3D space, 12�(G); limit set of group G, 62M(G); hyperbolic manifold
H3 ∪�(G)/G, 66�(G); discontinuity set of group G, 64δ-hyperbolic, δ-thinness, 109λ-Lemma, 362log(2k − 1) theorem on group actions,
272, 439MCG(R); mapping class group of
surface R, 91R(G); representation space of group G,
277Rdisc(G); discreteness locus = AH(G),
279Teich(R); Teichmüller space of surface
R, 87T(R),T(G); quasifuchsian space,
surface R, group G, 280
Abikoff, William, 198absolute measure of length, 7accidental parabolic, 198, 239Accola, Robert D. M., 65acylindrical manifold, 198, 239, 382Adams, Colin, 191, 252Adams, Scot, xviiiAgard, Steve, 96
Agol, Ian, xvii, xviii, 84, 114, 248, 252,293, 300, 358, 359, 363, 386, 389,396, 405, 422
Ahlfors, Lars, xiii, 17, 25, 42, 61, 77, 91,94, 96, 154, 186, 200, 204, 308,332, 336, 368, 456
Conjecture/Theorem, 184, 202, 295Finiteness Theorem, 64, 66, 115, 122,
192, 194, 234, 363Akiyoshi, Hirotaka, 144algebraic convergence, 219algebraic surface, 77Anderson, James, xvii, 81, 196, 209,
238, 284, 285, 329, 347, 349, 416,439
Andreev-Thurston Theorem, 10annulus, 25
modulus of, 336Anosov mappings of tori, 340anti-Möbius transformation, 1, 44Antonakoudis, Stergios, 381Aougab, Terik, 103area
as a function of topology, 187of disk, ball, 16of tube boundary, 104
arithmetic kleinian group, 401Arnoux, Pierre, 343Artin, Emil, 295Astala, Kari, 202
495
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
496 Index
atoroidal manifold, 382automatic group, 111automorphism
Dehn twist, 322iterated, 321
discrete group on S1, 395extention from S1 to disk, 395inner and outer, 283, 286, 353
of a 3-manifold, 370of fundamental group, 283pseudo-Anosov, 322reducible, 322
pseudo-Anosov, 322automorphism of a surface, 371, see also
Dehn twistAnosov maps on tori, 102finite order, 92
Nielsen Realization Problem, 92
B-groups, 309Baba, Shinpei, 415Baba, Shipei, xviiiball
circumference, 17volume, 17
ball, upper halfspace (UHS) modelsformulas for ball model, 30
Ballmann, Werner, 358baseball, 11Basmajian, Ara, xviii, 217Bass, Hyman, 383Beardon, Alan, 42, 133, 142, 145, 185,
199, 267, 444, 457Beltrami differential, 90, 288, 353, 368Beltrami, Eugenio, xvi
differential, 86for finitely generated kleinian
group, 183equation, 85, 86
Belyı functions, 117, 461bending
angle, 168
lamination, 170, 178, 179, 296, 302,366
lines, 172measure, 169
existence theorem, 178Benedetti, Riccardo, 250Bergeron, Nicolas, 386, 389Bers (analytic) boundary
geometric limits, 321, 346limit of iteration, 322locally connected case, 311
Bers slice, 306Bers (analytic) boundary, 310, 353extended, 308, 413
quasifuchsian locus, 413extened, 415
Bers, Lipman, 94, 145, 272, 281, 308,310, 336, 369
conjecture, see Density ConjectureBessiéres, Laurent, 395Besson, Gérard, 188, 395Bestvina, Mladen, 258, 259, 261, 354,
390Betti number/rank of H1(M3), 387Bianchi groups, 400bilipschitz map, 201billiards, 116Biringer, Ian, 239, 334Birman, Joan, 160, 342Bishop, Christopher, 201, 202, 334Bleiler, Steven, 34, 80, 192, 358Bobenko, Alexander, 10, 268Boileau, Michel, 199, 395, 402, 403Bólyai, János, xviBonahon, Francis, xvii, 26, 158, 160,
164, 165, 172, 178, 180, 195, 214,270, 282, 293, 295, 297, 298, 302,316, 385, 407
Criteria A and B, 292, 294, 310, 313,357
Bonk, Mario, 77Borromean rings, 398, 399, 401
approximation of complement, 397
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 497
boundarygroups, 309parallel embedded surface, 333
boundary componentcompressible (indecomposible), 355conformal, 73ideal, 76incompressible, 151, 382
bounded geometry, 303, 329Bounded Image Theorem, 382Bowditch
manifold constant, 176Bowditch, Brian, xviii, 24, 75, 110, 176,
184, 215, 279, 298, 300, 438Bowers, Philip, 10, 268brain cortex, 268branch cover, point, value, 68, 80branch locus, 70Brendle, Tara, 92Bridgeman, Martin, 174, 178, 198, 378Bridson, Martin R., 294Brin, Matthew, 358Brock, Jeffrey, xi, xviii, 238, 239, 293,
299, 300, 302, 303, 305, 314, 316,321, 322, 352, 353, 373, 390, 406,416
Bromberg, Kenneth, xviii, 50, 238, 239,287, 289, 293, 305, 310, 314, 373,406, 410, 411, 416
Brooks, Robert, xi, 117, 264, 440Brunner, Andrew, 401bumping, 289, 416, 418, 419
self-bumping, 289, 310Burger, Marc, 253, 334Buser, Peter, 133, 456Button, Jack, 393
Calegari, Daniel, 293, 300Callahan, David, 393Canary, Richard, xviii, 11, 115, 158,
160, 172, 173, 198, 201, 238, 279,282, 285, 286, 288, 292–295,
298–300, 303, 334, 337, 347, 349,354, 356–359, 363, 416, 439
Cannon Conjecture, 110Cannon, James, 36, 38, 42, 110, 111, 199Cannon-Thurston mappings, 327
a sufficient condition, 370degenerate Schottky groups, 332of kleinian groups, 330singly or doubly degenerated
quasifuchsian groups, 330Cao, Chun, 251Cao, Jian Guo, 212Carathéodory convergence, 233Casson, Andrew, 34, 80, 341, 386, 395CAT(−1), 294CAT(0), 393Cauchy–Riemann equations, 85Cayley graph, 108δ-thin, 109dual to polyhedra tessellation, 143geodesic, 108
Cayley–Hamilton identity, 425census of manifolds, 393cerebral cortex, 268character variety, 362Chavel, Isaac, 262Cheeger constant, 335Cheeger, Jeff, 34Choi, Young-Eun, 179Chow, Bennett, 395circle packing, 10, 264, 413
obtaining polyhedra, 268circles
euclidean and hyperbolic centers, 26circumference of a disk, 16closed manifold, 69collapsing laminations, 329collapsing map, 329, 339Collar Lemma, 133, 456combining groups, 152, 193commensurable groups, 196commensurator of a group, 196commutator, 2, 19
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
498 Index
subgroup, 67compact core, 180, 197, 281, 284, 291,
359, 435relative compact core, 181, 281, 291
ends of manifold, 291, 357companion knot/link, 396, 397complex length/distance, 431, 438, 441,
446between lines, 431
complex probabilities, 47complex projective structure, 412
grafting, 413monodromy (holonomy) group, 413
composition of Möbius t., 2compressible/incompressible
boundary component, 149surface, 151
compressible/incompressible surface,151
compressing curve, 83compressing disk, 151compression body, 83, 194, 354
embedding in S3, 397computer software, see also Snap,
SnapPi, OPTifor Bers slices, 415for cyclic loxodromic groups, 245
Conder, Marston, 252cone
angle/axis, 71, 254manifold, 255, 256, 406point, 69, 71, 76, 80
conformalaveraging, 440boundary, 12, 73groups, 107map, 1, 77metric, 85model, 8
congruence subgroup, 96conical limit point, 184, 198, 199conjugate
groups, 55
Möbius transformations, 2convergence, see also algebraic,
geometric, HausdorffCarathéodory, 233Gromov-Hausdorff c. of metric
spaces, 259of limit sets, 236of simple loops, 163type, 203
convex cocompact group, 177convex core, 115, 175, 177, 256, 279
bending measure, 178, 179boundary, 178, 303, 338, 363bounded embedded balls, 280bounded thickness, 176compact, 179in H2, 211maximal cusp, 319totally geodesic boundary, 377volume, 378
convex hull, 167bending measure, 169floor and dome, 169in H2, 419
Cooper, Daryl, xviii, 199, 262, 360, 402corner in manifold boundary, 420coset graph, 354Coulson, David, 400cover transformation, 68covering surface
branched, 68, 69normal, 68
Covering Theorem, 292coverings of surfaces/3-manifolds
normal coverings, 67regular coverings, 67Riemann surfaces, 67topological branched coverings, 393
Coxeter, H. M. S., 18critical exponent, 202cross ratio, 3, 5, 26, 35
and distances, angles, 25convergence, 54
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 499
cube complex/hyperplanes, 393Culler, Marc, 66, 246, 258, 277, 288,
303, 354, 399, 439curvature, see also under sectional
Gaussian, 47of arcs, 45of circle, 46of equidistant arc, 46of equidistant surface, 47of horocycles, 45sectional, 19
curve complex, 350arc complex, 352disk complex, 352pants complex, 352
cusp, see also under maximal cuspcusp cylinders/cusp tori Definitions,
125density, 288elimination, 359on deformation space boundary, 287,
313, 338paired punctures, solid pairing tube
Definition, 125rank of, 145solid cusp cylinders/solid cusp tori
Definitions, 125cyclic group, 56, 441Cylinder Theorem, 150, 156, 194cylindrical manifold, 198
Dahmani, Francois, 343De-Spiller, D. A., 200deck (=cover) transformation, 68deformation, see also under Teichmüller
spaceof kleinian groups, 281quasiconformal, 280, 287space, 276
interior of closure, 363space boundary
inclusiveness of groups, 305local connectivity, 416
degenerate groupcompression body, 304doubly, 304, 310, 373, 401partially, 313singly, 310, 313, 373
degenerate hexagon, 448degree of map to closed manifold, 393Dehn filling
exceptional slopes, 249on link complements, 398
Dehn surgery, 248, 404Dehn Surgery Theorem, 248, 404Dehn twist, 91, 339, 343
fixed point, 342iteration in T(R), 344surface automorphism, 339variation of length, 120
Dehn’s Lemma, xiiiDehn’s Lemma and Loop Theorem, 195
applications, 151equivariant, 150
Dehn-Nielsen-Baer Theorem, 353Delaunay triangulation, 206Density Conjecture/Theorem, 290, 305,
310dessins d’enfants, 117, 461developing map, 255, 413dihedral group, 60, 119dilatation, 85Dirichlet fundamental polyhedron, 135
generic:Jørgensen-Marden conjecture,208
discontinuity set �(G), 64discrete group, 55
with all real traces, 190discreteness locus, 279
in projective structure, 415disk
area, 17circumference, 17
diskbusting curves, 294, 357, 359, 405diskbusting link, 357divergence type, 203
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
500 Index
dodecahedral group, 60, 119dome over � ⊂ S2, 168
relation to geometry of �, 177Donaldson, Simon, 77Douady, Adrien, 154double horocycle, 147Double Limit Theorem, 304, 373double of a surface, 81doubling a manifold, 403drilling out simple geodesics, 404Drilling Theorem, 407Dumas, David, xi, 382, 412, 414, 415,
420Dunbar, William, 10, 252Dunfield, Nathan, 386, 390, 393, 396,
399, 424Duren, Peter, 233
Earle, Clifford, xvii, 94, 120, 141, 154,212, 218, 274, 309, 325, 344, 362
Earle–Marden coordinates, 274, 275earthquake, 210, 270Earthquake Theorem, 211edge cycle, 137edge relation, 135, 137Edmonds, Allan, 70Edmonds, Allan L., 393Efremovich, V., 199Ehrenpreis Conjecture, 87eigenvalues
geometrically finite/infinite, 333of a 2× 2 matrix, 4properties of λ1(M(G)), 335when λ0(M(G)) = 0, 334
electrification in geometric groups, 111elementary group, 56, 61, 94
all elements elliptic, 222elementary representation, 277elliptic transformation, 3
axis, 13maximal order in closed surface, 190
end reduction, 358
end/relative end of a manifold, 290, 291,294
case of a surface, 76compressible/incompressible, 293geometrically (in)finite, 291indecomposable, 293tame end, 291
ending lamination, 297, 298Conjecture/Theorem, 286, 290, 296,
300, 304, 373, 407definition, 301existence, 298
endpoint of geodesic, 12engulfing property, 358Epstein, David B. A., xviii, 27, 111, 142,
158, 160, 168–174, 177, 197, 206,211, 215, 217, 286, 398
equidistant curve/surface, 13, 46ergodicity, 204
and rigidity, 200unique, 166, 343
Eskin, Alex, 184essential cylinder (annulus), 150, 198
primitive, 349essential disk, 149ETH Zürich, xviiiEuclid, 6Euler characteristic, 69, 187Evans, Richard, xviii, 238, 239, 279, 293excess angle, 51exponential growth, 11extended (quasi)fuchsian group, 194extension ∂M(G)→M(G), 155, 240extension to S2 from �(G), 212extension to H3 of a univalent function,
50extremal length, 365
Farb, Benson, 92, 93, 111, 190, 343, 353Farey graph, 351Farey sequence, 49, 103, 104Farkas, Hershel, 77, 188
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 501
Fathi, Albert, 158, 163, 213, 316, 343,372, 456
Fatou, Pierre, 133Fay, John, 66Feighn, Mark, 197, 354Fenchel, Werner, 42, 44, 167, 208, 444Fenchel-Nielsen coordinates of
Teich(R), 94Ferguson, Helaman, 78, 379Fermat curve, 77Fermat’s Last Theorem, 96fibering over the circle, 374figure-8 knot, 190, 401filling/arational lamination, 166
filling pair, 167, 373finite group of Möbius transformations,
60finitely generated kleinian groups, 197finitely presented group, 74finiteness theorem
for cusps, 197for finite subgroups, 197
Fletcher, Alastair, 91foliation, see also measured foliation
(un)stable, 342Ford region/polygon/polyhedron, 138,
140Ford region/polyhedron, 245
finite-sided, 145generalization, 217
Ford, Lester R., 21, 60, 61, 119four-manifolds, 75Fox, Ralph, 295fractal, 64, 81, 114, 201fractional linear transformations, 1Frame, Michael, 401free group, 74, 81, 82
outer space, 354two generator, 335
free homotopy, 150Freedman, Michael, 293Fricke, Robert, 50fuchsian centers, 414
fuchsian group, 50, 62, 801st and 2nd kind, 81deformations, 88extended, 194finite index subgroups, 118finitely generated, 144geometric limits, 256least area, 189maximal, 189naming of, xviNielsen kernel, 211representation variety, 360triangle group, 61, 72, 98, 105, 189universal horodisks, 218
Fujii, Michihiko, 378function group, 65, 194fundamental group, 67fundamental polyhedron, 105, 135, 441
Dirichlet, 137Ford, 138
generalized, 138not locally finite, 142
Gabai, David, 133, 252, 283, 293, 300,341, 386, 395, 409, 454
with Meyerhoff and N. Thurston, 130,182, 437
Gallo, Daniel, 413, 415Gardiner. Fred, 96Gaster, Jonah, 376, 409Gauss, Johann Friedrich, xvGauss, map, hyperbolic, 50Gauss–Bonnet formula, 18, 179, 187,
188, 359gaussian curvature, 17gaussian integer, 105Gehring, Fred, 153, 155, 186, 200, 441Gelander, Tsachik, 253geodesic, 159, 264, 428
arclength, 32complex length, 431exiting sequence, 297length spectra of closed surfaces, 436
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
502 Index
penetration of horodisk, 190recurrent, 158self-intersecting, 217space of –s, 264unknotted, 409
geodesic lamination, 158, 419filling pair, 373maximal, 167measured, 161
projective, 162total angle measure of transverse
arc, 214uniquely ergodic, 167
minimal, 167realizable, 173, 296
geometric convergence, 185, 225, 257at Bers slice boundary, 344at quasifuchsian boundary, 323Benjamini-Schramm (BS)
convergence, 272, 390by renormalization, 271polyhedral, 226
geometric group theory, 108geometric intersection number, 101, 162,
339estimates, 213two measured laminations, 165
geometric structures, 394geometrically (in)finite end, 291geometrically finite groups, 144, 149
definitions, 145density on boundary, 305essential compactness, 144minimally parabolic, 284
Geometrization Conjecture/Theorem,xiv, 394, 396
Geometry Center, 111, 379Gilman, Jane, 56, 82, 115Goldman, William, 24, 31, 411, 412, 420Goodman, Oliver, 271, 400Gordon, Cameron, 396grafting, 410–412
2π-grafting, 411
Gray, Jeremy, xv, xviGreen’s formula, 19Green’s function, 204Green, Paul, 158, 160, 172, 173Greenberg, Leon, 41, 56, 65, 145, 147,
189, 196, 209, 235, 278, 369Gromov, Mikhail, 34, 110, 249, 259, 358
–’s Theorem, 263hyperbolicity, 109–111
a summary, 353norm, 263
Grothendieck, Alexandre, 117, 461group, see free, kleinian, quasifuchsian
etc.δ-hyperbolic, 110combination theory, 152, 193complex conjugate, 103containing only elliptics, 222Gromov hyperbolic, 110HNN extension, 293hyperbolic, 110indecomposable, 292inverse limit, 113Klein 4-group, 62LERF, 114marked, 277normalizer, 68presentation, 74profinite completion, 113relatively hyperbolic, 111residually finite, 114separable subgroup, 114word hyperbolic, 110
group properties, summary, 114Groves, Daniel, 386Guirardel, Vincent, 343Gunn, Charlie, 398, see Not KnotGuo, Ren, xviii, 45
Haefliger, André, 294Haglund, Frédéric, 386Haken manifold, 282, 383half-rotation, 425, 434, 444
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 503
Halpern, Naomi, 141, 218Hamenstädt, Ursula, 316, 351, 436Hamilton, Richard, 395handle, 76handlebody, 83, 340, 341harmonic (hyperbolic) maps, 154, 332Hartshorn, Kevin, 423Harvey, William, 65, 256, 350Hausdorff
convergencedefinition, 232of limit sets, 238, 239, 254
dimension, 201of limit sets, 333union of simple geodesics, 160
measure, 201Heard, Damian, 400Heegaard splitting, 84
Heegaard genus of M(G), 84splitting distance, 423
Hejhal, Dennis, 415, 418Hempel, John, 84, 112, 149, 152, 153,
195, 282, 382, 384, 394, 403Hersonsky, Sa’ar, 288, 303hexagonal, see also right hexagon
packing, 191, 268punctured torus, 103, 207torus, 190, 308, 309
Hidalgo, Rubén, 409hierarchy, 383Hilbert, David, 18
metric, 44Hildebrand, Martin, 393Hilden, H M, 399HNN-extension, 383Hocking, John, 65Hodgson, Craig, 41, 192, 248, 255, 358,
393, 400, 402, 405, 406holomorphic motion, 362holonomy group, 413holonomy map, 255Holt, John, 289, 310
homeomorphisms between manifolds,282
homologyS3, 390, 424basis, 101, 116group, 67Torelli group, 423
homotopy, 183, 205homotopy equivalence, 187, 281, 284
between manifolds, 282between surfaces, 282homeomorphisms, 282primitive shuffle, 285shuffle of rolodex, book pages, 285
horizon, 32horocycle, 14
double, 147foliation by –s, 214
horodisk, 14general form in �(G), 140in a torsion free fuchsian group, 218in simply connected region, 218penetration by geodesics, 190
horosphere and horoball, 14, 21, 123,127
maximal, 192penetration by planes, 190
Hubbard, John H., 10, 77, 91, 94hyperbolic
cone manifold, 255, 256, 406based on unknotted geodesic, 411deformations, 407
cube, 453Gauss map, 50geometry, 6group, 111harmonic maps, 154, 332law of (co)sines
for hexagons, 446for pentagons, 451for quadrilaterals, 452for triangles, 453
manifold, 66, see also under manifold
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
504 Index
boundary area, 350Bowditch constant, 176covering, 70cubulation, 393diameter bound, 262manifold double, 378Minsky Model Theorem, 299noncuspidal part, 294random choices, 423totally geodesic, 377volume bound, 262with corners, 420zero first homology, 397
metricsimply connected domain, 42
orbifold, 71, 256existence theorem, 402structure of singular set, 72
quadrilaterals, 454Ptolemy relation, 463
right hexagon, 446, 453, 455degenerate, 448, 450generic, 444
right triangle, 451space, 8transformation, 4trigonometry, 446
hyperbolic group, 109, 110hyperbolic knots, 396hyperbolic metric, 30
annulus, 95cylindrical (Fermi) coordinates, 457horocyclic coordinates, 457polar coordinates, 457punctured disk, 95solid angle, 13
hyperbolic plane, 8disk, upper half-plane (UHP) models,
8hyperbolic space, 8
ball, upper halfspace (UHS) models, 8polyhedra, 10
Hyperbolization Theorem, 110, 284,338, 348, 360, 371, 378, 382, 385
for surfaces, 79hyperboloid model, 37
imaginary length, 37light cone, 37timelike, lightlike, spacelike, 37
hyperelliptic involution, 100, 106
I-bundle, 81icosahedral group, 60, 119ideal
bigon, 162boundary component, 76line, 448, 449point, 14tetrahedra, 34, 191, 433, 437triangle, 7, 14triangulation, 269vertex, 7
incomplete hyperbolic metric, 254incompressible surface, 151, 382
doubly incompressible, 173indecomposable group, 151injectivity radius, 126, 397
positive lower bound, 303interval exchange transformations, 214invariant spiral, 4involution
conjugation by, 47hyperelliptic, 100, 106
irreducible manifold, 382irreducible representations, 278isometric circle, 20, 22
excess, 51isometric plane, 20isoparametric inequality, 111, 262isothermal coordinates, 85isotopy, 183, 205
mapping class group, 91of metrics, 314
Ito, Kentaro, 418Ivanov, Nikolai, 93, 343, 351
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 505
Jørgensen, Troels, xiv, xvii, 33, 47, 49,51, 56, 105, 134, 143–145, 219,225, 231, 234, 235, 237, 245, 284,308, 310, 319, 425, 434, 441
complex probabilities, 308inequality, 56, 57, 105, 127, 143, 220,
223, 225, 226, 229, 441cases of equality, 105
new parabolics, 237Jaco, William, 84, 149, 152, 363, 382,
384, 394Jacobi identity, 441jet, 50Johannson, Klaus, 240, 281, 394Jones, Gareth, 117, 188Jones, Peter, 201, 202, 334Jungreis, Douglas, 341, 386, 395
Kahn, Jeremy, xvii, 87, 378, 389Kamishima, Yoshinobu, 419, 420Kapovich, Michael, 123, 184, 208, 248,
253, 259, 277, 333, 363, 385, 402,413, 415, 420
Kapovich, Misha, xviiiKeen, Linda, 103, 133Kellerhals, Ruth, 252Kent IV, Richard P., 382, 408Kerckhoff, Steven, xviii, 92, 164, 165,
211, 214, 248, 255, 314, 325, 341,344, 346, 356, 365, 402, 404–406
Kiikka, Maire, 105Klarreich, Erica, 329, 351Klein, Felix, 7
bottle, 403model, 37surface, 78
Klein, Peter, 219Klein–Maskit combination theory, 152,
193Kleineidam, Gero, 304, 354–356Kleiner, Bruce, 395kleinian group, 62
arithmetic, 401
convergence, 231determined by its traces, 435doubly degenerate, 374finitely generated, 197higher dimensional, 253infinitely generated, 66, 326, 421naming of, xvipresentation, 74quaternion representation, 42two-generator, 49, 363, 435
Knopf, Dan, 395knot
complement, 105hyperbolic structure, 396
figure-8, 190, 375longitude and meridian, 397Seifert surface, 397
Knotted Wye, 379knotted/unknotted geodesic, 409Kojima, Sadayoshi, xviii, 255, 404, 413Komori, Yohei, 308, 415Korkmaz, Mustafa, 351Koundouros, Stelios, 397Kra, Irwin, 94, 188, 281, 369Kulkarni, Ravi, 70, 197
Labourie, Francois, 68Lackenby, Marc, 249, 360Lakes of Wada, 65Lakic, Nikola, 96lamination, see also under geodesic
lamination(un)stable, 342, 372ending, 301minimal, 161
Lamping, John, 11laplacian (hyperbolic), 332, 333Lascurain, Antonio, 105lattice, 99Laudenbach, Francois, 158, 163, 213,
316, 343, 372Le Calvez, Patrice, 92Le, Thang, 390
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
506 Index
Lecuire, Cyril, 178, 354Lee, Youn, 401Leeb, Bernhard, 402, 403Lehner, Joseph, 57, 96Lehto, Olli, 91, 107Leininger, Chris, 114LERF, (all f.g. subgroups separable),
112, 114Leung, Naichung, 182level k congruence subgroup, 96Levy, Silvio, xi, xviiiLi, Peter, 155Li, Tao, 84Lickorish, Raymond, 397Lie product, 426Lieninger, Chris, 196lifting to a matrix group, 66limit set, 63, see also under conical limit
setconvergence, 236Hausdorff dimension, 201Hausdorff distance, 238locally connected, 331tangents, 107
line geometry, 425link
complement, 105Dehn surgery along, 249diskbusting, 358, 359indecomposable, 397
linked/unlinked geodesics, 409Liouville measure of geodesics, 26Liu, Yi, 262, 389Lobachevsky, function, 35Lobachevsky, Nikolai Ivanovich, xvilocal connectivity, 287
quasifuchsian discreteness locus, 287of limit sets, 327
Long, Darren D., 196longitude and meridian, 397Loop Theorem, see Dehn’s LemmaLott, John, 395loxodromic
curve, 4transformation, 3
axis, 13Lozano, M T, 399Lubotzky, Alexander, 253Luecke, John, 396Luo, Feng, 351Lyndon, Roger C., 354
Möbius strip, 264Möbius transformation, 1
axis, 13, 427, 434composition, 2composition of reflections, 27convergence, 53eigenvalues and eigenvectors, 4extension to 3-space, 6half-rotation, 425images of horizontal lines, 121in ≥ 3 dimensions (Liouville’s
theorem), 28normalized, 2normalized matrix representation, 2square roots, 429standard forms, 4
Maclachlan, Colin, 383, 401Magid, Aaron, 287magnetic resonance imaging, 268Magnus, Wilhelm, 50, 98Maher, Joseph, 111, 423Maillot, Sylvain, 395Maloni, Sara, xviiiMané, Ricardo, 362Mandelbrot, Benoît, 81, 201manifold, see also under hyperbolic
manifoldaspherical, 386boundary topology, 152containing knots/unknots, 409from face pairing of polyhedra, 424geometrically atoroidal, 383graph of manifolds, 354Haken, 383
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 507
incompressible, 383boundary incompressible, 383
irreducible, 383pared, 383random choice, 424toroidal=homotopically t., 383
manifold vs orbifold, 67manifolds
higher dimensional, 253Manning, Jason, 386mapping class group, 111, 372
5-punctured sphere, 343action on Thurston boundary, 316classification of elements, 341closed hyperbolic 3-manifold, 183definition, 91exceptional cases, 92extended, 92extension to M(G), 364finite index subgroup, 93finite subgroups, 341not realizable by homeos, 92puncture fixing subgroup, 93random walk, 424rigidity, 187torsion free, finite index normal
subgroup, 341mapping torus, 374Marden, Albert, 27, 80, 82, 123, 141,
152, 168–171, 174, 177, 195, 197,211, 218, 234, 235, 237, 274, 280,281, 286, 313, 325, 343, 344, 364,365, 369, 377, 413, 415
Conjecture, see Tameness ConjectureIsomorphism (or Rigidity) Theorem,
182, 183, 346Margalit, Dan, 93, 190, 353Margulis, Grigori, 200, 400
constant, 134Markov identity and conjecture, 23, 24Markovic, Vlad, 155
Markovic, Vladimir, xvii, xviii, 87, 91,92, 110, 118, 154, 168–170, 174,177, 208, 212, 389
Marshall, Timothy, 252Martin, Gaven, 252, 441Maskit, Bernard, xiv, 65, 82, 107, 152,
157, 185, 193, 195, 198, 199, 281,290, 309, 310, 313, 329, 338, 369,409
Planarity Theorem, 82, 115, 153Masur, Howard, xvii, xviii, 111, 116,
184, 214, 300, 316, 350, 353, 356,364, 463
domain, 354–356Matelski, Peter, 440Mathematical Sciences Research
Institute, 78matrix group from kleinian group, 66Matsuzaki, Katsuhiko, 73, 202, 204, 240maximal cusp, 287, 289, 313, 338
density of –s, 314maximal lamination, 166Maxwell, Delle, see Not KnotMcCullough, Darryl, 180, 195, 197, 282,
285, 286, 355McMullen, Curtis, xi, 96, 154, 215, 254,
279, 288, 303, 308, 314, 329, 336,338, 363, 373, 377, 381, 400, 416
McReynolds, David, 114McShane, Gregory
Mcshane identity, 438measured
foliation, 165from interval exchange, 214horocyclic, 214quadratic differentials, 366
lamination, 161arational/filling, 166by sequence of lengths, 164finite, 163length, 164quadratic differentials, 364sequence convergence, 163
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
508 Index
uniquely ergodic, 166, 167Meeks III, William H., 150meromorphic
function, 117, 411, 419locally injective, 412
lamination, 419quadratic differential, 461
Mess, Geoffrey, 75, 197Meyerhoff, Robert, 129, 130, 133, 134,
182, 249, 251, 252, 283, 409, 454Miller, Andrew, 180, 195, 355Milley, Peter, 133, 252Milnor, John, xvi, 35, 395minimal lamination, 161, 166minimally parabolic, 284Minkowski space, 38
light cone, 40timelike, lightlike, spacelike, 40
Minsky Model Theorem, 299Minsky, Yair, xi, xviii, 111, 173, 293,
298–300, 303, 308, 328, 334, 339,350, 353, 365, 375, 410, 416
Mirzakhani, Maryam, 160geodesic length formula, 438
Mitra, Sudab, 154Miyachi, Hideki, 308Mj, Mahan, xvii, 330, 375Möbius transformation
orientation reversing, see anti-Möbiusmodular group, 96
=mapping class group, 92extended, 92, 94Farey sequence, 104
modular transformation, 100moduli space, 94
compactifications, 94definition, 93manifold cover, 94of a 3-manifold, 370triangulation and compactification,
215modulus
of annulus, 336
of circular quadrilateral, 265monodromy group, 413Montel’s Theorem, 53Montesinos, José Maria, 394, 399Moore, R.L., 328, 332Mordell Conjecture, 77Morgan, John, 258, 385, 395Mosher, Lee, 343Mostow, George, xiii, 187, 200, 377
Rigidity Theorem, 182, 186, 283, 378history, 200
Mozes, Shasar, 253MSRI, 78multicurve, 162, 412
intersection numbers, 162Mumford, David, xvii, 49, 81, 104, 115,
142, 253, 257, 287Munkres, James, 153Munzner, Tamara, 11, 66, 96Myers, Robert, 358, 405, 408
Namazi, Hossein, 305Nash, John, 76navigation, 4nearest point retraction, 168, 169, 175,
419negative curvature, see also pinching
and hyperboliccharacterizations, 17discrete, 17of groups, 109, 111
nerve, 265Neumann, Walter, 191, 271, 400new parabolics, 237Nicholls, Peter, 202, 204, 205Nielsen Realization Problem, 92, 211,
341Nielsen, Jakob, 167, 208
kernel, 211transformation, 335
non-Euclidean geometry, xvnormal subgroup, 68normalizer of a subgroup, 68
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 509
Not Knot, 66, 96, 398number theory, 96
octagon, hyperbolic, 427octahedral group, 60, 119Ohshika, Ken’ichi, xvii, xviii, 110, 238,
258, 293, 295, 305, 323, 337, 422Ol’shanskiı, A. Yu., 111OPTi, 308orbifold, 67, 70, 402
cover, 73euclidean, 120minimum volume, 252spherical, 119theorem, 402
Orbifold Theorem, 403orbital counting function, 203ordinary set �(G), 64oriented lines, 431, 446orthogonal projection, 11, 460Osin, Denis, 343Otal, Jean-Pierre, 164, 178, 258, 259,
355, 356, 374, 385, 407, 409, 420outer circles, 20, 21Outer space, 354
page shuffling, see rolodexpaired punctures, 147
joining, 274opening up, 284
pants, 455all medium sized, 272cuff lengths, 456decomposition, 165, 269, 288, 304,
421pants complex, 352Papakyriakopoulos, Christos, xiiiparabolic group, 98, 190
discrete extension, 102horosphere and horoball, 14intrinsic horosphere euclidian metric,
124least (translation) length, 126
parabolic transformation, 3accidental, 198associated geometric structures, 145formulas for, 427new parabolics, 237
parallel loops, 337pared manifold, 382Parker, John, xi, 180, 468, 471Parker–Series bending formulas, 468,
471Patterson, Samuel J., 202, 205Patterson-Sullivan measure on limit sets,
204Peano curve
equivariant construction, 327, 332Penner, Howard, xiPenner, Robert C., 217, 343, 463pentagon, 451, 454, 460Perelman, Grigori, xiv
Geometerization Theorem, 385Petersen–Morley Theorem, 445Petronio, Carlo, 142, 207, 248, 250, 270Picard group, 105Pignataro, Thea, 105pinching
curvature bounds, 358estimate, 288, 336limiting process, 253, 287, 289, 304,
309, 312, 319, 337loops, 313, 323Theorem (Ohshika), 337
Pirelli, Peter, 11planar
covering surface, 81pentagon, 451, 455quadrilateral, 454Riemann surface, 82right hexagon, 455
pleated surfaces, 171, 173uniform injectivity, 173
pleating locus, 172plumbing coordinates, 274PNC manifolds, 358, 360
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
510 Index
Poénaru, Valentin, 158, 163, 213, 316,343, 372
Poincaré, Henri, xiii, xvi, 12, 38, 135Conjecture, 12, 84, 382dodecahedral space, 12Polyhedron Theorem, 142series, 205
point of approximation, see also underconical limit set
Poisson integral formula, 332Pollicott, Mark, 369polygon, hyperbolic, 7polyhedral
convergence, 226deformation, 36group, 60, 116, 118surface, 17, 76
polyhedronhyperbolic, 10volume, 36
Pommerenke, Christian, 201, 212, 233,329, 331
Porti, Joan, 248, 250, 395, 402, 403Prasad, Gopal, 182, 187presentation, 74
length, 262primitive curve, 271primitive group element, 59profinite completion, 118projective model, 37projective structure
discreteness locus, 415, 419extended Bers slice, 414quasifuchsian locus, 414Thurston coordinates, 420
properly discontinuous, 56, 64, 78Przeworski, Andrew, 133, 252pseudo-Anosov mappings, 342, 372pseudosphere, 18puncture, 76, 78punctured torus, 103, 335, 439, 441
group, 103, 144, 313, 319, 439, 441,465
hexagonal, 103, 207Purcell, Jessica S., 397
quadratic differential, 90, 205, 364Abelian differential, 463bundle over T(R), 415, 419singular euclidean metric, 463
quadratic differentials, 90quadratic forms, 96quadrilateral
circular, 264marked, 265planar with two right angles, 454with three right angles, 452
quasiconformaldefinition, 85deformation, 86, 463deformation space, 280extension �(G) to S2, 213metric definition, 85, 200
quasifuchsian, see also degenerate groupdeformation space, 305, 373group, 155, 194, 305
illustration, 311, 312quasifuchsian space, 308
Bers slices, 308Earle slice, 309Maskit slice, 308nonlocal connectivity of closure, 416
quasiisometry, 155, 199, 200quasisymmetric boundary mapping, 91quaternions, 42, 436
RAAG: right-angled Artin group, 392Rafi, Kasra, 184, 351ramification, see branchrank of cusp, 145rank two subgroups, 272Rao, Ramana, 11Ratcliffe, John, 35, 73, 191, 252real R-trees, 258, 259, 362real projective structure, 411realizable lamination, 173, 296
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 511
recurrent geodesic, 158recurrent ray, 185, 199reducible, see also irreduciblereducible automorphism, 341reducible group, 95
representation, 277Rees, Mary, 165reflection
in a point, 44in a sphere or plane, 1, 25, 29, 44
regular exhaustion, 290regular set �(G), 64Reid, Alan, 196, 271, 383, 401Reimann, Hans Martin, 154relatively hyperbolic group, 111
with respect to subgroup, 111relator, 74representation variety, 276
character variety, 362discreteness locus, 279, 289, 416, 418fuchsian groups, 360local coordinates, 179quasiconformal deformation space,
280residually finite, 112residually finite group, 112, 114retraction
in hyperbolically convex set, 197nearest point, 168, 169, 175, 419of H3 to line, 11
rhumb line, 4Ricci flow, 395Riemann Mapping Theorem, 77Riemann surface, 75
(integral) grafting, 412Belyi functions, 117, 461built from pants, 455canonical hyperbolic triangulations,
215compact bordered, 81conformal embedding in R3, 76cut into polygons, 463Ehrenpreis Conjecture, 87
from equilateral triangles, 461from ideal triangles, 269from interval exchange, 214from pentagons, 460genus 2, 107isometric embeddings in R17,R51, 76length of closed geodesics, 133marked, 88polygon decompositions, 217projective structure, 415spine, 216translation surface, 463
Riemann–Hurwitz formula, 69, 118riemannian 3-manifolds
pinched negative sectional curvature,386
right triangle, 450, 454rigidity
3-manifolds with boundary, 183mapping class group, 187of homotopies, 182of homotopy equivalences, 282quasiconformal, 281topological, 303
Riley, Robert, xiv, 143, 375, 400, 401Rips, Eliahu, 109Rivin, Igor, 10, 268Rodin, Burt, 266Roeder, Roland K.W., 10, 268Rogness, Jonathan, xiRolfsen, Dale, 399rolodex, 285, 347, 349Royden, Halsey, 93Rüedy, Reto, 76
Sad, Ricardo, 362Sageev, Michah, 386, 390Sakai, Tsuyoshi, 404Sakuma, Makoto, 144Saric, Dragomir, 118Sario, Leo, 77, 82satellite knot/link, 396, 397Scannell, Kevin P., 412
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
512 Index
Schafer, James, 117Schläfli formula, 36Schleimer, Saul, xviii, 351, 352Schneps, Leila, 461Schoen Conjecture, 155Schottky group, 81, 87, 114, 115, 144,
152, 195, 257boundary cusp, 289classical, 82degenerated, 331dimension of limit set, 201illustration, 307simply degenerate, 332
Schroeder, Viktor, 358Schupp, Paul E., 354Schwartz, Rick, 439Schwarz Lemma, 456schwarzian derivative, 50, 413schwarzian differential equations, 413Scott, Peter, xiv, xviii, 74, 75, 112, 180,
194, 295, 385, 394, 405Scott–Shalen theorem, 276sectional curvature, 19, 182
pinched, 358Seifert Conjecture, 386, 395Seifert fiber space, 394, 395Seifert–Weber dodecahedral space, 139Selberg’s Lemma, 57, 73, 148, 184, 197,
359separable subgroup, 112, 114Seppälä, Mika, 436Series, Caroline, xi, 49, 81, 104, 115,
120, 142, 160, 179, 180, 257, 287,468, 471
Shalen, Peter, 74, 133, 180, 197, 246,258, 277, 288, 303, 363, 394, 439
Sharp, Richard, 369Shiga, Hiroshige, 414Shimizu, Hideo, 134Shinnar, Meir, 134short geodesics, 127, 438
drilling out, 407shrinkwrapping, 294
shuffle, 285, 347, 349Siegel, Carl Ludwig, 57, 96Sierpinski gasket (carpet), 378, 379simplicial volume, 264simultaneous uniformization, 157, 305,
306Singerman, David, 117singular fiber, 394singular set of an orbifold, 71, 402Skinning Lemma, 323, 376, 408skinning map, 376Skora, Richard, 259Slodkowski, Zbigniew, 362Smillie, John, 463SnapPea/SnapPy, 270, 399solenoid, 118solid torus, 83
longitude and meridian, 397Soma, Teruhiko, xvii, 294, 323, 408, 422Sorvali, Tuomas, 436soul, 212Souto, Juan, xviii, 238, 239, 280, 293,
294, 305, 334, 354–356, 397, 405,409
sphere at infinity, 12sphere theorem, 382spherical manifold, 12spine, 135spinning, 269Springborn, Boris, 10Springer, George, 77square root of Möbius t., 429stabilizer, 55standard form of Möbius t., 4Stephenson, Kenneth, xi, 10, 267, 268stereographic projection, 1, 28, 36
B3 → UHS., 29Stillwell, John, xviStong, Robert, 70Storm, Peter, 188, 378, 396, 403Strebel, Kurt, 89, 343, 364, 365strong convergence, 239strong stability, 280
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 513
Strong Torus Theorem, 385structural stability of groups, 363Struik, Dirk, 17, 19Sturm, Jacob, 134subgroup separable, 114Sugawa, Toshiyuki, 308, 415Sullivan, Dennis, 110, 170, 197, 204,
205, 266, 334, 362, 363group stability, 363K-theorem, floor to dome, 170Rigidity Theorem, 86, 183, 303, 308,
310Sun, Hongbin, 390surface area and volume, 262surface automorphims
finite order, 341surface automorphisms
Dehn twistsiterates, 316, 325
pseudo-Anosov, 327, 341, 372axis, 342fixed points, 372iterates, 322, 325rank and abelian group, 372
reducible, 322, 341Surface Subgroup Conjecture/Theorem,
387counting immersed surfaces, 388
Swarup, Ananda, 180symmetry lines, 442systole, 103, 437
Tam, Luem-Fai, 155tame
end, 291, 386manifold, 291, 292, 422, see also
Ahlfors’ Conjecture, Bonahon’sCriteria, untameness
Tameness Conjecture (Theorem), 360Tameness Conjecture/Theorem, 238,
239, 289, 293, 294, 299, 334, 363Tan, Ser Peow, 413, 419, 420
tangent bundle of a hyperbolic surface,33
tangents to limit sets, 107Tanigawa, Harumi, 414Taylor, Edward, 201, 334Teichmüller lemma, 87Teichmüller mapping, 89
extremal, 90Teichmüller modular group
= mapping class group, 92Teichmüller space
Bers (analytic) boundary, 310Bers slice, 307biholomorphic automorphisms, 94bounded orbits, 381comparison Bers and Thurston
boundaries, 316complex structure, 94definitions, 88dimension, 89geodesic rigidity, 215geodesics, 90global complex analytic coordinates,
94global real analytic coordinates, 94higher Teichmüller space, 369isometries, 94metric, 89natural tessellation, 215pseudo-Anosov action, 373quasi-isometric rigidity, 184ray, 364relative hyperbolicity, 111surface with cone points, 369Thurston (geometric) boundary, 316
convergent sequences, 315pseudo-Anosov fixed points, 373
Teichmüller, Oswald, 87tetrahedral group, 60, 119tetrahedron, flattened, 270tetrahedron, ideal, 433, 437
thinness, 36volume, 36
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
514 Index
Theorema Egregium, 18thick/thin decomposition, 134, 173, 192Thickstun, Thomas, 358thin part, see thick/thin decompositionThurston, Nathaniel, 130, 182, 283, 409,
454Thurston, William, xiv, xvii, 50, 111,
112, 154, 163, 167, 169, 187, 191,199, 215, 246, 248, 249, 253, 268,290, 292, 293, 295, 297, 298, 314,325, 334, 339, 341, 343, 344, 346,358, 372, 373, 375, 377, 379, 382,386, 393–396, 398, 401–403, 419,424
orbifold/reflection trick, 420Compactness Theorem, 240, 261, 279,
280, 338coordinates, 420earthquakes, 210, 211geometric finiteness, 149, 192Gluing Theorem, 381Hyperbolization Theorem, 385pleated surfaces, 173thick/thin, 173
Tihomirova, E., 199topological rigidity, 182, 282Torelli group
homology spheres, 424torsion-free, 55, 66Torstensen, Anna, 252torus, 99, 190, see also punctured torus
hexagonal, 190, 308, 309knot, 396marked, 100slope of simple loop, 100square, 103
Torus Theorem, 405totally geodesic boundary, 179, 377trace
–s determine group, 435and Dehn twist, 343calculations for cyclic groups, 32definition, 2
identities, 23, 32, 47, 50signed, 24
train tracks, 366switch condition, 367weighted, 367
Tranah, David, xviiitriangle
area, 7, 19, 459area and side length, 27group, 61, 98uniform thinness, 15
tripod, 258Tschantz, Steven, 252tubular neighborhood
of geodesic, 13, 26, 123, 130, 133volume/area, 104
of systole, 437universal, 134
Tucker, Thomas, 295Tukia, Pekka, 154, 183, 184, 186, 209twisted I-bundle over Klein bottle, 403type preserving, 88, 277
UHP, 28, see upper half-planeuniform injectivity, 269Uniformization Theorem, 77uniformization, simultaneous, 157, 305uniformly perfect set, 43, 272uniquely ergodic, 166, 343universal
ball, 127ball ⊂M(G), 127constants, 127cover, 67
PSL(2,R), 68elementary neighborhood, 127horoball/horosphere, 127horodisk, 190
extended form in �(G), 141hyperbolic solenoid, 118isolation of cone axes, 127tubular neighborhoods, 127, 134
universe, curvature of, 12
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
Index 515
University of Minnesota, xviii, 379University of Warwick, xviiiUnknottedness Theorem, 409untameness, 296
van Kampen’s Theorem, 151, 383Van Vleck, Edward, 27vector space of 2× 2 matrices, 436Virtual Domination
Conjecture/Theorem, 390Virtual Fibering Conjecture/Theorem,
387Virtual Haken Conjecture/Theorem, 386visual angle, 32visual sphere, 12Vogtmann, Karen, 354volume
3-manifold minimums, 2513-orbifold minimums, 251estimated by thick part, 104geometrically finite, 149higher dimensions, 252manifold bound, 262of ball, 16of convex core, 378of hyperbolic manifolds, 245, 396of maximal horoball, 192of polyhedra, 36of tetrahedra, 36of tubes, 104simplicial (of a manifold), 264well ordering, 251
Voronoi diagram, 206Vuorinen, Matti, 28
Wada, Masaaki, 47, 144, 308, 415, 439Waldhausen, Friedhelm, xiv, 84, 152,
186, 282, 383, 386Wan, Tom, 182Wang, Hsien-chung, 134, 253Waterman, Peter, 82, 115Weeks manifold, 401Weeks, Jeffrey, xvii, 12, 41, 207, 270,
393, 399Weil-Petersson metric, 352, 353Weiss, Hartmut, 256Whitehead link, 252, 401Whitehead, George, 282Whitten, W C, 399Wielenberg, Norbert, 105, 143, 149, 400,
401wild embedding, 295Wiles, Andrew, 96Wise, Daniel T., xvii, 389Wolf, Michael, 353, 412Wolpert, Scott, 120, 353, 436word-hyperbolic, 109wormhole, 379wrapping around a loop, 417Wright, David, xi, xvii, 49, 81, 104, 115,
142, 257, 287, 378
Yamada, Akira, 217Yamashita, Yasushi, 144, 308Yau, Shing-Tung, 150Yoccoz, Jean-Christophe, 343Young, Gail S., 65
Zhu, Xiaodong, 302
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 DimensionsAlbert MardenIndexMore information
top related