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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

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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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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






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