Volume 1 (1998)

Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
All Volumes
 
About this Series
Ethics Statement
Purchase Printed Copies
Author Index
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
 
MSP Books and Monographs
Other MSP Publications
A brief survey of the deformation theory of Kleinian groups

James W Anderson

Geometry & Topology Monographs 1 (1998) 23–49
DOI: 10.2140/gtm.1998.1.23
Bibliography
1 W Abikoff, B Maskit, Geometric decompositions of Kleinian groups, Amer. J. Math. 99 (1977) 687 MR0480992
2 L V Ahlfors, Lectures on quasiconformal mappings, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software (1987) MR883205
3 L V Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964) 413 MR0167618
4 L V Ahlfors, The structure of a finitely generated Kleinian group, Acta Math. 122 (1969) 1 MR0237782
5 L Ahlfors, L Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. $(2)$ 72 (1960) 385 MR0115006
6 J W Anderson, R D Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math. 126 (1996) 205 MR1411128
7 J W Anderson, R D Canary, Cores of hyperbolic 3–manifolds and limits of Kleinian groups, Amer. J. Math. 118 (1996) 745 MR1400058
8 J W Anderson, R D Canary, Cores of hyperbolic 3–manifolds and limits of Kleinian groups II, J. London Math. Soc. $(2)$ 61 (2000) 489 MR1760675
9 J W Anderson, R D Canary, M Culler, P B Shalen, Free Kleinian groups and volumes of hyperbolic 3–manifolds, J. Differential Geom. 43 (1996) 738 MR1412683
10 J W Anderson, R D Canary, D McCullough, The topology of deformation spaces of Kleinian groups, Ann. of Math. $(2)$ 152 (2000) 693 MR1815699
11 J W Anderson, R D Canary, D McCullough, The topology of deformation spaces of Kleinian groups, Ann. of Math. $(2)$ 152 (2000) 693 MR1815699
12 A Basmajian, S A Wolpert, Hyperbolic 3-manifolds with nonintersecting closed geodesics, Geom. Dedicata 97 (2003) 251 MR2003700
13 R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer (1992) MR1219310
14 L Bers, On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math. $(2)$ 91 (1970) 570 MR0297992
15 L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94 MR0111834
16 L Bers, Spaces of Kleinian groups, from: "Several Complex Variables I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970)", Springer (1970) 9 MR0271333
17 C J Bishop, P W Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997) 1 MR1484767
18 F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. $(2)$ 124 (1986) 71 MR847953
19 F Bonahon, A Schläfli-type formula for convex cores of hyperbolic 3–manifolds, J. Differential Geom. 50 (1998) 25 MR1678473
20 F Bonahon, Variations of the boundary geometry of 3–dimensional hyperbolic convex cores, J. Differential Geom. 50 (1998) 1 MR1678469
21 F Bonahon, J P Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988) 255 MR931188
22 B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245 MR1218098
23 B H Bowditch, Spaces of geometrically finite representations, Ann. Acad. Sci. Fenn. Math. 23 (1998) 389 MR1642122
24 R Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. (1979) 11 MR556580
25 J F Brock, Iteration of mapping classes on a Bers slice: examples of algebraic and geometric limits of hyperbolic 3–manifolds, from: "Lipa's legacy (New York, 1995)", Contemp. Math. 211, Amer. Math. Soc. (1997) 81 MR1476982
26 J F Brock, Iteration of mapping classes and limits of hyperbolic 3–manifolds, PhD thesis, UC Berkeley (1997)
27 R D Canary, Algebraic convergence of Schottky groups, Trans. Amer. Math. Soc. 337 (1993) 235 MR1137257
28 R D Canary, Covering theorems for hyperbolic 3–manifolds, from: "Low-dimensional topology (Knoxville, TN, 1992)", Conf. Proc. Lecture Notes Geom. Topology, III, Int. Press, Cambridge, MA (1994) 21 MR1316167
29 R D Canary, Ends of hyperbolic 3–manifolds, J. Amer. Math. Soc. 6 (1993) 1 MR1166330
30 R D Canary, Geometrically tame hyperbolic 3–manifolds, from: "Differential geometry: Riemannian geometry (Los Angeles, CA, 1990)", Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 99 MR1216615
31 R D Canary, On the Laplacian and the geometry of hyperbolic 3–manifolds, J. Differential Geom. 36 (1992) 349 MR1180387
32 R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)", London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3 MR903850
33 R D Canary, D McCullough, Homotopy equivalences of 3–manifolds and deformation theory of Kleinian groups, Mem. Amer. Math. Soc. 172 (2004) MR2096234
34 R D Canary, Y N Minsky, On limits of tame hyperbolic 3–manifolds, J. Differential Geom. 43 (1996) 1 MR1424418
35 R D Canary, Y N Minsky, E C Taylor, Spectral theory, Hausdorff dimension and the topology of hyperbolic 3–manifolds, J. Geom. Anal. 9 (1999) 17 MR1760718
36 R D Canary, E C Taylor, Hausdorff dimension and limits of Kleinian groups, Geom. Funct. Anal. 9 (1999) 283 MR1692482
37 V Chuckrow, On Schottky groups with applications to kleinian groups, Ann. of Math. $(2)$ 88 (1968) 47 MR0227403
38 T D Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups, PhD thesis, University of Michigan (1996)
39 M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237 MR881270
40 M Culler, P B Shalen, Varieties of group representations and splittings of 3–manifolds, Ann. of Math. $(2)$ 117 (1983) 109 MR683804
41 M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537 MR735339
42 C E Fan, Injectivity radius bounds in hyperbolic convex cores, PhD thesis, University of Michigan (1997)
43 C E Fan, Injectivity radius bounds in hyperbolic I-bundle convex cores, preprint
44 C E Fan, J Jorgenson, Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds, from: "Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999)", Contemp. Math. 290, Amer. Math. Soc. (2001) 57 MR1868468
45 F W Gehring, J Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. $(2)$ 6 (1973) 504 MR0324028
46 L Greenberg, Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962) 85 MR0141731
47 M Gromov, Hyperbolic manifolds (according to Thurston and J\o rgensen), from: "Bourbaki Seminar, Vol. 1979/80", Lecture Notes in Math. 842, Springer (1981) 40 MR636516
48 J Hempel, 3–Manifolds, Ann. of Math. Studies 86, Princeton University Press (1976) MR0415619
49 W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society (1980) MR565450
50 W Jaco, P B Shalen, Seifert fibered spaces in 3–manifolds, from: "Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977)", Academic Press (1979) 91 MR537728
51 K Johannson, Homotopy equivalences of 3–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer (1979) MR551744
52 T Jørgensen, On cyclic groups of Möbius transformations, Math. Scand. 33 (1973) MR0348103
53 T Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976) 739 MR0427627
54 T Jørgensen, P Klein, Algebraic convergence of finitely generated Kleinian groups, Quart. J. Math. Oxford Ser. $(2)$ 33 (1982) 325 MR668178
55 T Jørgensen, A Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990) 47 MR1060898
56 L Keen, C Series, Pleating coordinates for the Teichmüller space of a punctured torus, Bull. Amer. Math. Soc. $($N.S.$)$ 26 (1992) 141 MR1110439
57 L Keen, C Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993) 719 MR1241870
58 L Keen, C Series, Pleating invariants for punctured torus groups, Topology 43 (2004) 447 MR2052972
59 L Keen, C Series, The Riley slice of Schottky space, Proc. London Math. Soc. $(3)$ 69 (1994) 72 MR1272421
60 S P Kerckhoff, W P Thurston, Noncontinuity of the action of the modular group at Bers' boundary of Teichmüller space, Invent. Math. 100 (1990) 25 MR1037141
61 I Kra, On lifting Kleinian groups to $\mathrm{SL}(2,\mathbb{C})$, from: "Differential geometry and complex analysis" (editors I Chavel, H Farkas), Springer (1985) 181 MR780044
62 I Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972) 53 MR0306485
63 O Lehto, K I Virtanen, Quasiconformal mappings in the plane, Die Grundlehren der mathematischen Wissenschaften 126, Springer (1973) MR0344463
64 A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. $(2)$ 99 (1974) 383 MR0349992
65 A Marden, Geometric relations between homeomorphic Riemann surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 3 (1980) 1001 MR585180
66 B Maskit, On boundaries of Teichmüller spaces and on Kleinian groups II, Ann. of Math. $(2)$ 91 (1970) 607 MR0297993
67 B Maskit, A characterization of Schottky groups, J. Analyse Math. 19 (1967) 227 MR0220929
68 B Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften 287, Springer (1988) MR959135
69 B Maskit, Parabolic elements in Kleinian groups, Ann. of Math. $(2)$ 117 (1983) 659 MR701259
70 B Maskit, Self-maps on Kleinian groups, Amer. J. Math. 93 (1971) 840 MR0291453
71 K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, Oxford University Press (1998) MR1638795
72 D McCullough, Compact submanifolds of 3–manifolds with boundary, Quart. J. Math. Oxford Ser. $(2)$ 37 (1986) 299 MR854628
73 D McCullough, A Miller, G A Swarup, Uniqueness of cores of noncompact 3–manifolds, J. London Math. Soc. $(2)$ 32 (1985) 548 MR825931
74 C McMullen, Cusps are dense, Ann. of Math. $(2)$ 133 (1991) 217 MR1087348
75 C T McMullen, Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999) 471 MR1726737
76 Y N Minsky, The classification of punctured-torus groups, Ann. of Math. $(2)$ 149 (1999) 559 MR1689341
77 J W Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, from: "The Smith conjecture (New York, 1979)", Pure Appl. Math. 112, Academic Press (1984) 37 MR758464
78 J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures I, Ann. of Math. $(2)$ 120 (1984) 401 MR769158
79 J W Morgan, P B Shalen, Degenerations of hyperbolic structures II: Measured laminations in 3–manifolds, Ann. of Math. $(2)$ 127 (1988) 403 MR932305
80 J W Morgan, P B Shalen, Degenerations of hyperbolic structures III: Actions of 3–manifold groups on trees and Thurston's compactness theorem, Ann. of Math. $(2)$ 127 (1988) 457 MR942518
81 G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press (1973) MR0385004
82 K Ohshika, Divergent sequences of Kleinian groups, from: "The Epstein birthday schrift", Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 419 MR1668324
83 K Ohshika, Ending laminations and boundaries for deformation spaces of Kleinian groups, J. London Math. Soc. $(2)$ 42 (1990) 111 MR1078179
84 K Ohshika, Geometric behaviour of Kleinian groups on boundaries for deformation spaces, Quart. J. Math. Oxford Ser. $(2)$ 43 (1992) 97 MR1150472
85 K Ohshika, Geometrically finite Kleinian groups and parabolic elements, Proc. Edinburgh Math. Soc. $(2)$ 41 (1998) 141 MR1604337
86 K Ohshika, Kleinian groups which are limits of geometrically finite groups, Mem. Amer. Math. Soc. 177 (2005) MR2154090
87 K Ohshika, Strong convergence of Kleinian groups and Carathéodory convergence of domains of discontinuity, Math. Proc. Cambridge Philos. Soc. 112 (1992) 297 MR1171166
88 J P Otal, Sur la dégénérescence des groupes de Schottky, Duke Math. J. 74 (1994) 777 MR1277954
89 J P Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3, Astérisque (1996) MR1402300
90 J P Otal, F Paulin, Géométrie hyperbolique et groupes kleiniens, in preparation
91 G Prasad, Strong rigidity of $\mathbb{Q}$–rank 1 lattices, Invent. Math. 21 (1973) 255 MR0385005
92 D Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982) 99 MR684247
93 G P Scott, Compact submanifolds of 3–manifolds, J. London Math. Soc. $(2)$ 7 (1973) 246 MR0326737
94 P B Shalen, Representations of 3–manifold groups and applications in topology, from: "Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986)", Amer. Math. Soc. (1987) 607 MR934262
95 D Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, from: "Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978)", Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 465 MR624833
96 D Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984) 259 MR766265
97 D Sullivan, Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985) 243 MR806415
98 E C Taylor, Geometric finiteness and the convergence of Kleinian groups, Comm. Anal. Geom. 5 (1997) 497 MR1487726
99 W P Thurston, Hyperbolic structures on 3–manifolds I: Deformation of acylindrical manifolds, Ann. of Math. $(2)$ 124 (1986) 203 MR855294
100 W P Thurston, Hyperbolic structures on $3$–manifolds II: Surface groups and $3$–manifolds which fiber over the circle, preprint
101 W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 6 (1982) 357 MR648524
102 W P Thurston, The geometry and topology of $3$–manifolds, Lecture notes, Princeton University (1978)
103 D J Wright, The shape of the boundary of Maskit's embedding of the Teichmüller space of once punctured tori, preprint