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, $2$-generator Kleinian groups, and volumes of hyperbolic $3$-manifolds
HTML articles powered by AMS MathViewer

by Marc Culler and Peter B. Shalen

J. Amer. Math. Soc. 5 (1992), 231-288

DOI: https://doi.org/10.1090/S0894-0347-1992-1135928-4

PDF | Request permission

References

Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 167618, DOI 10.2307/2373173
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, DOI 10.1017/S0004972700004275
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
K. Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 3-4, 243–261. MR 512399, DOI 10.1007/BF01902361
Richard D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 6 (1993), no. 1, 1–35. MR 1166330, DOI 10.1090/S0894-0347-1993-1166330-8

Covering theorems for hyperbolic

-manifolds

Vicki Chuckrow, On Schottky groups with applications to kleinian groups, Ann. of Math. (2) 88 (1968), 47–61. MR 227403, DOI 10.2307/1970555
H. S. M. Coxeter, Non-Euclidean geometry, Mathematical Expositions, No. 2, University of Toronto Press, Toronto, Ont., 1957. 3rd ed. MR 0087965
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, DOI 10.2307/2006973
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, DOI 10.1515/9783110849455
F. W. Gehring and G. J. Martin, Inequalities for Möbius transformations and discrete groups, J. Reine Angew. Math. 418 (1991), 31–76. MR 1111201
John Hempel, $3$-Manifolds, Annals of Mathematics Studies, No. 86, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. MR 0415619
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, DOI 10.1090/memo/0220
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
Troels Jørgensen, Closed geodesics on Riemann surfaces, Proc. Amer. Math. Soc. 72 (1978), no. 1, 140–142. MR 503548, DOI 10.1090/S0002-9939-1978-0503548-2
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
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
Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383–462. MR 349992, DOI 10.2307/1971059
Curt McMullen, Cusps are dense, Ann. of Math. (2) 133 (1991), no. 1, 217–247. MR 1087348, DOI 10.2307/2944328
Robert Meyerhoff, Sphere-packing and volume in hyperbolic $3$-space, Comment. Math. Helv. 61 (1986), no. 2, 271–278. MR 856090, DOI 10.1007/BF02621915
Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
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
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, DOI 10.1090/S0002-9939-1990-1013981-6
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, DOI 10.1090/S0002-9947-1992-1156298-8
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
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

Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension

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, DOI 10.1007/BF02392543

The geometry and topology of

-manifolds

On Kleinian groups and

-manifolds of Euler characteristic zero

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, DOI 10.1017/CBO9780511609596
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594