Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds

Culler, Marc; Shalen, Peter B.

doi:10.1090/S0894-0347-1992-1135928-4

Paradoxical decompositions, -generator Kleinian groups, and volumes of hyperbolic -manifolds

Authors: Marc Culler and Peter B. Shalen
Journal: J. Amer. Math. Soc. 5 (1992), 231-288
MSC: Primary 57M50; Secondary 30F40
DOI: https://doi.org/10.1090/S0894-0347-1992-1135928-4
MathSciNet review: 1135928
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[A] Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 167618, https://doi.org/10.2307/2373173
[BaS] Gilbert Baumslag and Peter B. Shalen, Groups whose three-generator subgroups are free, Bull. Austral. Math. Soc. 40 (1989), no. 2, 163–174. MR 1012825, https://doi.org/10.1017/S0004972700004275
[Be] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
[Bo] Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, https://doi.org/10.2307/1971388
[Bör] K. Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 3-4, 243–261. MR 512399, https://doi.org/10.1007/BF01902361
[Ca1] Richard D. Canary, Ends of hyperbolic 3-manifolds, J. Amer. Math. Soc. 6 (1993), no. 1, 1–35. MR 1166330, https://doi.org/10.1090/S0894-0347-1993-1166330-8
[Ca2] -, Covering theorems for hyperbolic -manifolds, preprint.
[Ch] Vicki Chuckrow, On Schottky groups with applications to kleinian groups, Ann. of Math. (2) 88 (1968), 47–61. MR 227403, https://doi.org/10.2307/1970555
[Co] H. S. M. Coxeter, Non-Euclidean geometry, Mathematical Expositions, no. 2, University of Toronto Press, Toronto, Ont., 1957. 3rd ed. MR 0087965
[CuS] Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109–146. MR 683804, https://doi.org/10.2307/2006973
[F] Werner Fenchel, Elementary geometry in hyperbolic space, De Gruyter Studies in Mathematics, vol. 11, Walter de Gruyter & Co., Berlin, 1989. With an editorial by Heinz Bauer. MR 1004006
[GeM] F. W. Gehring and G. J. Martin, Inequalities for Möbius transformations and discrete groups, J. Reine Angew. Math. 418 (1991), 31–76. MR 1111201
[He] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
[JaS] William H. Jaco and Peter B. Shalen, Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, https://doi.org/10.1090/memo/0220
[Joh] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
[Jør] Troels Jørgensen, Closed geodesics on Riemann surfaces, Proc. Amer. Math. Soc. 72 (1978), no. 1, 140–142. MR 503548, https://doi.org/10.1090/S0002-9939-1978-0503548-2
[K] A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842
[Ma1] A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 405–422 (Russian, with English summary). MR 0003420
[Mar] Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383–462. MR 349992, https://doi.org/10.2307/1971059
[McM] Curt McMullen, Cusps are dense, Ann. of Math. (2) 133 (1991), no. 1, 217–247. MR 1087348, https://doi.org/10.2307/2944328
[Me] Robert Meyerhoff, Sphere-packing and volume in hyperbolic 3-space, Comment. Math. Helv. 61 (1986), no. 2, 271–278. MR 856090, https://doi.org/10.1007/BF02621915
[N] Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575
[P] S. J. Patterson, Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 281–323. MR 903855
[ScS] G. P. Scott and G. A. Swarup, Geometric finiteness of certain Kleinian groups, Proc. Amer. Math. Soc. 109 (1990), no. 3, 765–768. MR 1013981, https://doi.org/10.1090/S0002-9939-1990-1013981-6
[ShW] Peter B. Shalen and Philip Wagreich, Growth rates, 𝑍_{𝑝}-homology, and volumes of hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 895–917. MR 1156298, https://doi.org/10.1090/S0002-9947-1992-1156298-8
[Su1] Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586
[Su2] Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, 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., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
[Su3] -, Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension 2, Geometry Symposium Utrecht 1980, Lecture Notes in Math., vol. 894, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 127-144.
[Su4] Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), no. 3-4, 243–260. MR 806415, https://doi.org/10.1007/BF02392543
[Th] W. Thurston, The geometry and topology of -manifolds, Photocopied notes, Princeton Univ., Princeton, NJ, 1978.
[Tuc] T. Tucker, On Kleinian groups and -manifolds of Euler characteristic zero, unpublished.
[Wag] Stan Wagon, The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski. MR 803509
[Wa1] Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, https://doi.org/10.2307/1970594