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Hyperbolic Manifold H3 Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Index More information Index 2π Lemma, 359 Agol, Ian, xvii, xviii, 84, 114, 248, 252, H2, H3; hyperbolic 2D, 3D space, 12 293, 300, 358, 359, 363, 386, 389, (G); limit set of group G, 62 396, 405, 422 M(G); hyperbolic manifold Ahlfors, Lars, xiii, 17, 25, 42, 61, 77, 91, H3 ∪ (G)/G, 66 94, 96, 154, 186, 200, 204, 308, (G); discontinuity set of group G, 64 332, 336, 368, 456 δ-hyperbolic, δ-thinness, 109 Conjecture/Theorem, 184, 202, 295 λ-Lemma, 362 Finiteness Theorem, 64, 66, 115, 122, log(2k − 1) theorem on group actions, 192, 194, 234, 363 272, 439 Akiyoshi, Hirotaka, 144 algebraic convergence, 219 MCG(R); mapping class group of surface R, 91 algebraic surface, 77 Anderson, James, xvii, 81, 196, 209, R(G); representation space of group G, 238, 284, 285, 329, 347, 349, 416, 277 439 R (G); discreteness locus = AH(G), disc Andreev-Thurston Theorem, 10 279 annulus, 25 Teich(R); Teichmüller space of surface modulus of, 336 R, 87 Anosov mappings of tori, 340 T(R), T(G); quasifuchsian space, anti-Möbius transformation, 1, 44 surface R, group G, 280 Antonakoudis, Stergios, 381 Aougab, Terik, 103 Abikoff, William, 198 area absolute measure of length, 7 as a function of topology, 187 accidental parabolic, 198, 239 of disk, ball, 16 Accola, Robert D. M., 65 of tube boundary, 104 acylindrical manifold, 198, 239, 382 arithmetic kleinian group, 401 Adams, Colin, 191, 252 Arnoux, Pierre, 343 Adams, Scot, xviii Artin, Emil, 295 Agard, Steve, 96 Astala, Kari, 202 495 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Index More information 496 Index atoroidal manifold, 382 lamination, 170, 178, 179, 296, 302, automatic group, 111 366 automorphism lines, 172 Dehn twist, 322 measure, 169 iterated, 321 existence theorem, 178 discrete group on S1, 395 Benedetti, Riccardo, 250 extention from S1 to disk, 395 Bergeron, Nicolas, 386, 389 inner and outer, 283, 286, 353 Bers (analytic) boundary of a 3-manifold, 370 geometric limits, 321, 346 of fundamental group, 283 limit of iteration, 322 pseudo-Anosov, 322 locally connected case, 311 reducible, 322 Bers slice, 306 pseudo-Anosov, 322 Bers (analytic) boundary, 310, 353 automorphism of a surface, 371, see also extended, 308, 413 Dehn twist quasifuchsian locus, 413 extened, 415 Anosov maps on tori, 102 Bers, Lipman, 94, 145, 272, 281, 308, finite order, 92 310, 336, 369 Nielsen Realization Problem, 92 conjecture, see Density Conjecture Bessiéres, Laurent, 395 B-groups, 309 Besson, Gérard, 188, 395 Baba, Shinpei, 415 Bestvina, Mladen, 258, 259, 261, 354, Baba, Shipei, xviii 390 ball Betti number/rank of H (M3), 387 circumference, 17 1 Bianchi groups, 400 volume, 17 bilipschitz map, 201 ball, upper halfspace (UHS) models billiards, 116 formulas for ball model, 30 Biringer, Ian, 239, 334 Ballmann, Werner, 358 Birman, Joan, 160, 342 baseball, 11 Bishop, Christopher, 201, 202, 334 Basmajian, Ara, xviii, 217 Bleiler, Steven, 34, 80, 192, 358 Bass, Hyman, 383 Bobenko, Alexander, 10, 268 Beardon, Alan, 42, 133, 142, 145, 185, Boileau, Michel, 199, 395, 402, 403 199, 267, 444, 457 Bólyai, János, xvi Beltrami differential, 90, 288, 353, 368 Bonahon, Francis, xvii, 26, 158, 160, Beltrami, Eugenio, xvi 164, 165, 172, 178, 180, 195, 214, differential, 86 270, 282, 293, 295, 297, 298, 302, for finitely generated kleinian 316, 385, 407 group, 183 Criteria A and B, 292, 294, 310, 313, equation, 85, 86 357 Belyˇı functions, 117, 461 Bonk, Mario, 77 bending Borromean rings, 398, 399, 401 angle, 168 approximation of complement, 397 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Index More information Index 497 boundary 298–300, 303, 334, 337, 347, 349, groups, 309 354, 356–359, 363, 416, 439 parallel embedded surface, 333 Cannon Conjecture, 110 boundary component Cannon, James, 36, 38, 42, 110, 111, 199 compressible (indecomposible), 355 Cannon-Thurston mappings, 327 conformal, 73 a sufficient condition, 370 ideal, 76 degenerate Schottky groups, 332 incompressible, 151, 382 of kleinian groups, 330 bounded geometry, 303, 329 singly or doubly degenerated Bounded Image Theorem, 382 quasifuchsian groups, 330 Bowditch Cao, Chun, 251 manifold constant, 176 Cao, Jian Guo, 212 Carathéodory convergence, 233 Bowditch, Brian, xviii, 24, 75, 110, 176, Casson, Andrew, 34, 80, 341, 386, 395 184, 215, 279, 298, 300, 438 CAT(−1), 294 Bowers, Philip, 10, 268 CAT(0), 393 brain cortex, 268 Cauchy–Riemann equations, 85 branch cover, point, value, 68, 80 Cayley graph, 108 branch locus, 70 δ-thin, 109 Brendle, Tara, 92 dual to polyhedra tessellation, 143 Bridgeman, Martin, 174, 178, 198, 378 geodesic, 108 Bridson, Martin R., 294 Cayley–Hamilton identity, 425 Brin, Matthew, 358 census of manifolds, 393 Brock, Jeffrey, xi, xviii, 238, 239, 293, cerebral cortex, 268 299, 300, 302, 303, 305, 314, 316, character variety, 362 321, 322, 352, 353, 373, 390, 406, Chavel, Isaac, 262 416 Cheeger constant, 335 Bromberg, Kenneth, xviii, 50, 238, 239, Cheeger, Jeff, 34 287, 289, 293, 305, 310, 314, 373, Choi, Young-Eun, 179 406, 410, 411, 416 Chow, Bennett, 395 Brooks, Robert, xi, 117, 264, 440 circle packing, 10, 264, 413 Brunner, Andrew, 401 obtaining polyhedra, 268 bumping, 289, 416, 418, 419 circles self-bumping, 289, 310 euclidean and hyperbolic centers, 26 Burger, Marc, 253, 334 circumference of a disk, 16 Buser, Peter, 133, 456 closed manifold, 69 Button, Jack, 393 collapsing laminations, 329 collapsing map, 329, 339 Calegari, Daniel, 293, 300 Collar Lemma, 133, 456 Callahan, David, 393 combining groups, 152, 193 Canary, Richard, xviii, 11, 115, 158, commensurable groups, 196 160, 172, 173, 198, 201, 238, 279, commensurator of a group, 196 282, 285, 286, 288, 292–295, commutator, 2, 19 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Index More information 498 Index subgroup, 67 Möbius transformations, 2 compact core, 180, 197, 281, 284, 291, convergence, see also algebraic, 359, 435 geometric, Hausdorff relative compact core, 181, 281, 291 Carathéodory, 233 ends of manifold, 291, 357 Gromov-Hausdorff c. of metric companion knot/link, 396, 397 spaces, 259 complex length/distance, 431, 438, 441, of limit sets, 236 446 of simple loops, 163 between lines, 431 type, 203 complex probabilities, 47 convex cocompact group, 177 complex projective structure, 412 convex core, 115, 175, 177, 256, 279 grafting, 413 bending measure, 178, 179 monodromy (holonomy) group, 413 boundary, 178, 303, 338, 363 composition of Möbius t., 2 bounded embedded balls, 280 compressible/incompressible bounded thickness, 176 boundary component, 149 compact, 179 surface, 151 in H2, 211 compressible/incompressible surface, maximal cusp, 319 151 totally geodesic boundary, 377 compressing curve, 83 volume, 378 compressing disk, 151 convex hull, 167 compression body, 83, 194, 354 bending measure, 169 embedding in S3, 397 floor and dome, 169 computer software, see also Snap, in H2, 419 SnapPi, OPTi Cooper, Daryl, xviii, 199, 262, 360, 402 for Bers slices, 415 corner in manifold boundary, 420 for cyclic loxodromic groups, 245 coset graph, 354 Conder, Marston, 252 Coulson, David, 400 cone cover transformation, 68 angle/axis, 71, 254 covering surface manifold, 255, 256, 406 branched, 68, 69 point, 69, 71, 76, 80 normal, 68 conformal Covering Theorem, 292 averaging, 440 coverings of surfaces/3-manifolds boundary, 12, 73 normal coverings, 67 groups, 107 regular coverings, 67 map, 1, 77 Riemann surfaces, 67 metric, 85 topological branched coverings, 393 model, 8 Coxeter, H. M. S., 18 congruence subgroup, 96 critical exponent, 202 conical limit point, 184, 198, 199 cross ratio, 3, 5, 26, 35 conjugate and distances, angles, 25 groups, 55 convergence, 54 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11674-0 - Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions Albert Marden Index More information Index 499 cube complex/hyperplanes, 393 degenerate group Culler, Marc, 66, 246, 258, 277, 288, compression body, 304 303, 354, 399, 439 doubly, 304, 310, 373, 401 curvature, see also under sectional partially, 313 Gaussian, 47 singly, 310, 313, 373 of arcs, 45 degenerate hexagon, 448 of circle, 46 degree of map to closed manifold, 393 of equidistant arc, 46 Dehn filling of equidistant surface, 47 exceptional slopes, 249 of horocycles, 45 on link complements, 398 sectional, 19 Dehn surgery, 248, 404 curve complex, 350 Dehn Surgery Theorem, 248, 404 arc complex, 352 Dehn twist, 91, 339, 343 disk complex, 352 fixed point, 342 pants complex, 352 iteration in T(R), 344 cusp, see also under maximal cusp surface automorphism, 339 cusp cylinders/cusp tori Definitions, variation of length, 120 125 Dehn’s Lemma, xiii density, 288 Dehn’s Lemma and Loop Theorem, 195 elimination, 359 applications, 151 on deformation space boundary, 287, equivariant, 150 313, 338 Dehn-Nielsen-Baer Theorem, 353 paired punctures, solid pairing tube Delaunay triangulation, 206 Definition, 125 Density Conjecture/Theorem, 290, 305, rank of, 145 310 solid cusp cylinders/solid cusp tori dessins d’enfants, 117, 461 Definitions, 125 developing map, 255,
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