Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Growth rates, $ Z\sb p$-homology, and volumes of hyperbolic $ 3$-manifolds


Authors: Peter B. Shalen and Philip Wagreich
Journal: Trans. Amer. Math. Soc. 331 (1992), 895-917
MSC: Primary 57M05; Secondary 20F05, 57M07, 57N10
DOI: https://doi.org/10.1090/S0002-9947-1992-1156298-8
MathSciNet review: 1156298
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if $ M$ is a closed orientable irreducible $ 3$-manifold and $ n$ is a nonnegative integer, and if $ {H_1}(M,{\mathbb{Z}_p})$ has rank $ \geq n + 2$ for some prime $ p$, then every $ n$-generator subgroup of $ {\pi _1}\,(M)$ has infinite index in $ {\pi _1}\,(M)$, and is in fact contained in infinitely many finite-index subgroups of $ {\pi _1}\,(M)$. This result is used to estimate the growth rates of the fundamental group of a $ 3$-manifold in terms of the rank of the $ {\mathbb{Z}_p}$-homology. In particular it is used to show that the fundamental group of any closed hyperbolic $ 3$-manifold has uniformly exponential growth, in the sense that there is a lower bound for the exponential growth rate that depends only on the manifold and not on the choice of a finite generating set. The result also gives volume estimates for hyperbolic $ 3$-manifolds with enough $ {\mathbb{Z}_p}$-homology, and a sufficient condition for an irreducible $ 3$-manifold to be almost sufficiently large.


References [Enhancements On Off] (What's this?)

  • [BaS] G. Baumslag and P. B. Shalen, Groups whose $ 3$-generator subgroups are free, Bull. Austral. Math. Soc. 40 (1989), 163-174. MR 1012825 (90h:20041)
  • [Be] A. F. Beardon, The geometry of discrete groups, Graduate Texts in Math., vol. 91, Springer-Verlag, 1983. MR 698777 (85d:22026)
  • [Bö] K. Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), 243-261. MR 512399 (80h:52014)
  • [C] J. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. MR 758901 (86j:20032)
  • [CS] M. Culler and P. B. Shalen, Paradoxical decompositions, Margulis numbers and volumes of hyperbolic $ 3$-manifolds, Preprint, Univ. of Illinois at Chicago.
  • [EM] B. Evans and L. Moser, Solvable fundamental groups of compact $ 3$-manifolds, Trans. Amer. Math. Soc. 168 (1972), 189-210. MR 0301742 (46:897)
  • [F] P. Fatou, Fonctions automorphes, Vol. 2, Théorie des Fonctions Algébriques (P. E. Appell and E. Goursat, Eds.), Gauthiers-Villars, Paris, 1930, pp. 158-160.
  • [Gr] M. Gromov, Structures métriques pour les variétés Riemanniennes, Fernand-Nathan, Paris.
  • [He] J. Hempel, $ 3$-manifolds, Ann. of Math. Studies, no. 86, Princeton Univ. Press, 1976. MR 0415619 (54:3702)
  • [JaS] W. H. Jaco and P. B. Shalen, Seifert fibered spaces in $ 3$-manifolds, Mem. Amer. Math. Soc. 21, no. 220(1979). MR 539411 (81c:57010)
  • [Jo] K. Johannson, Homotopy equivalences of $ 3$-manifolds with boundaries, Lecture Notes in Math., vol. 761, Springer-Verlag, Berlin, 1979. MR 551744 (82c:57005)
  • [K] A. G. Kurosh, Theory of groups, vol. II, Chelsea, 1960. MR 0109842 (22:727)
  • [L.] A. Lubotzky, Group presentation, $ p$-adic analytic groups and lattices in $ {\text{SL}}_2(\mathbb{C})$, Ann. of Math. 118 (1983), 115-130. MR 707163 (85i:22017)
  • [Ma1] A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Math. Sb.], (N.S.) 8 (50) (1940), 405-422. MR 0003420 (2:216d)
  • [Mar] G. A. Margulis, Arithmeticity of non-uniform lattices, Funkcional. Anal. i Priložen. 7 (1973), 88-89. MR 0330314 (48:8651)
  • [MeeSY] W. Meeks, L. Simon and S. T. Yau, Embedded minimal surfaces, Ann. of Math. 116 (1982), 621-659. MR 678484 (84f:53053)
  • [Mes1] G. Mess, Centers of $ 3$-manifold groups and groups which are coarse quasiisometric to planes, Preprint, Univ. of Calif., Los Angeles, 1990.
  • [Mes2] -, Finite covers and a theorem of Lubotzky, Preprint, Univ. of Calif., Los Angeles.
  • [Mey1] R. Meyerhoff, A lower bound for the volume of hyperbolic manifolds, Canad. J. Math. 39 (1987), 1038-1056. MR 918586 (88k:57049)
  • [Mey2] -, Sphere-packing and volume in hyperbolic $ 3$-space, Comment. Math. Helv. 61 (1986), 271-278. MR 856090 (88e:52023)
  • [Mi1] J. Milnor, A unique decomposition theorem for $ 3$-manifolds, Amer. J. Math. 84 (1962), 1-7. MR 0142125 (25:5518)
  • [Mi2] -, A note on curvature and fundamental groups, J. Differential Geometry 2 (1968), 1-7. MR 0232311 (38:636)
  • [P] W. Parry, A sharper Tits alternative for $ 3$-manifold groups, Preprint, Eastern Michigan Univ. MR 1194795 (93j:57002)
  • [Sc1] P. Scott, Finitely generated $ 3$-manifold groups are finitely presented, J. London Math. Soc. 2 (1973), 437-440. MR 0380763 (52:1660)
  • [Sc2] -, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980), 241-277. MR 564473 (81f:57006)
  • [Sc3] -, There are no fake Seifert fibered spaces with infinite $ {\pi _1}$, Ann. of Math. 117 (1983), 35-70. MR 683801 (84c:57008)
  • [Sh] P. B. Shalen, A torus theorem for regular branched coverings of $ {S^3}$, Michigan Math. J. 28 (1981), 347-358. MR 629367 (83d:57012)
  • [St1] J. Stallings, On the loop theorem, Ann. of Math. 72 (1960), 12-19. MR 0121796 (22:12526)
  • [St2] -, Homology and lower central series of groups, J. Algebra 2 (1965), 170-181. MR 0175956 (31:232)
  • [Th] W. P. Thurston, Geometry and toplogy of $ 3$-manifolds, Photocopied notes, Princeton Univ., 1978.
  • [Tuc] T. Tucker, On Kleinian groups and $ 3$-manifolds of Euler characteristic zero, Unpublished.
  • [Tur] V. G. Turaev, Nilpotent homotopy types of closed $ 3$-manifolds, (Topology, Leningrad, 1982), Lecture Noes in Math., vol. 1060, Springer-Verlag, 1984. MR 770255 (86i:57017)
  • [Wa] P. Wagreich, Singularities of complex surfaces with solvable local fundamental group, Topology 11 (1972), 51-72. MR 0285536 (44:2754)
  • [We] B. A. F. Wehrfritz, Infinite linear groups, Ergebnisse der Math. Grenzgebiete 76, Springer-Verlag, 1973. MR 0335656 (49:436)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M05, 20F05, 57M07, 57N10

Retrieve articles in all journals with MSC: 57M05, 20F05, 57M07, 57N10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1156298-8
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society