Growth rates, -homology, and volumes of hyperbolic -manifolds

Authors:
Peter B. Shalen and Philip Wagreich

Journal:
Trans. Amer. Math. Soc. **331** (1992), 895-917

MSC:
Primary 57M05; Secondary 20F05, 57M07, 57N10

MathSciNet review:
1156298

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if is a closed orientable irreducible -manifold and is a nonnegative integer, and if has rank for some prime , then every -generator subgroup of has infinite index in , and is in fact contained in infinitely many finite-index subgroups of . This result is used to estimate the growth rates of the fundamental group of a -manifold in terms of the rank of the -homology. In particular it is used to show that the fundamental group of any closed hyperbolic -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 -manifolds with enough -homology, and a sufficient condition for an irreducible -manifold to be almost sufficiently large.

**[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**, 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****[Bö]**K. Böröczky,*Packing of spheres in spaces of constant curvature*, Acta Math. Acad. Sci. Hungar.**32**(1978), no. 3-4, 243–261. MR**512399**, 10.1007/BF01902361**[C]**James W. Cannon,*The combinatorial structure of cocompact discrete hyperbolic groups*, Geom. Dedicata**16**(1984), no. 2, 123–148. MR**758901**, 10.1007/BF00146825**[CS]**M. Culler and P. B. Shalen,*Paradoxical decompositions, Margulis numbers and volumes of hyperbolic*-*manifolds*, Preprint, Univ. of Illinois at Chicago.**[EM]**Benny Evans and Louise Moser,*Solvable fundamental groups of compact 3-manifolds*, Trans. Amer. Math. Soc.**168**(1972), 189–210. MR**0301742**, 10.1090/S0002-9947-1972-0301742-6**[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]**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**, 10.1090/memo/0220**[Jo]**Klaus Johannson,*Homotopy equivalences of 3-manifolds with boundaries*, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR**551744****[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****[L.]**Alexander Lubotzky,*Group presentation, 𝑝-adic analytic groups and lattices in 𝑆𝐿₂(𝐶)*, Ann. of Math. (2)**118**(1983), no. 1, 115–130. MR**707163**, 10.2307/2006956**[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]**G. A. Margulis,*Arithmeticity of nonuniform lattices*, Funkcional. Anal. i Priložen.**7**(1973), no. 3, 88–89 (Russian). MR**0330314****[MeeSY]**William Meeks III, Leon Simon, and Shing Tung Yau,*Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature*, Ann. of Math. (2)**116**(1982), no. 3, 621–659. MR**678484**, 10.2307/2007026**[Mes1]**G. Mess,*Centers of*-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]**Robert Meyerhoff,*A lower bound for the volume of hyperbolic 3-manifolds*, Canad. J. Math.**39**(1987), no. 5, 1038–1056. MR**918586**, 10.4153/CJM-1987-053-6**[Mey2]**Robert Meyerhoff,*Sphere-packing and volume in hyperbolic 3-space*, Comment. Math. Helv.**61**(1986), no. 2, 271–278. MR**856090**, 10.1007/BF02621915**[Mi1]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****[Mi2]**J. Milnor,*A note on curvature and fundamental group*, J. Differential Geometry**2**(1968), 1–7. MR**0232311****[P]**Walter Parry,*A sharper Tits alternative for 3-manifold groups*, Israel J. Math.**77**(1992), no. 3, 265–271. MR**1194795**, 10.1007/BF02773691**[Sc1]**G. P. Scott,*Finitely generated 3-manifold groups are finitely presented*, J. London Math. Soc. (2)**6**(1973), 437–440. MR**0380763****[Sc2]**Peter Scott,*A new proof of the annulus and torus theorems*, Amer. J. Math.**102**(1980), no. 2, 241–277. MR**564473**, 10.2307/2374238**[Sc3]**Peter Scott,*There are no fake Seifert fibre spaces with infinite 𝜋₁*, Ann. of Math. (2)**117**(1983), no. 1, 35–70. MR**683801**, 10.2307/2006970**[Sh]**Peter B. Shalen,*A torus theorem for regular branched covers of 𝑆³*, Michigan Math. J.**28**(1981), no. 3, 347–358. MR**629367****[St1]**John Stallings,*On the loop theorem*, Ann. of Math. (2)**72**(1960), 12–19. MR**0121796****[St2]**John Stallings,*Homology and central series of groups*, J. Algebra**2**(1965), 170–181. MR**0175956****[Th]**W. P. Thurston,*Geometry and toplogy of*-*manifolds*, Photocopied notes, Princeton Univ., 1978.**[Tuc]**T. Tucker,*On Kleinian groups and*-*manifolds of Euler characteristic zero*, Unpublished.**[Tur]**V. G. Turaev,*Nilpotent homotopy types of closed 3-manifolds*, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 355–366. MR**770255**, 10.1007/BFb0099951**[Wa]**Philip Wagreich,*Singularities of complex surfaces with solvable local fundamental group*, Topology**11**(1971), 51–72. MR**0285536****[We]**B. A. F. Wehrfritz,*Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76. MR**0335656**

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:
http://dx.doi.org/10.1090/S0002-9947-1992-1156298-8

Article copyright:
© Copyright 1992
American Mathematical Society