Duke Mathematical Journal

Covering spaces of 3-orbifolds

Marc Lackenby

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Abstract

Let O be a compact orientable 3-orbifold with nonempty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {Oi} with linear growth of mod p homology for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of Oi grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. First, the fundamental group of O has at least exponential subgroup growth. Second, the covers {Oi} have positive Heegaard gradient. Third, we use the existence of this tower of covers to show that a group-theoretic conjecture of Lubotzky and Zelmanov implies that O has a large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with nonempty singular locus has large fundamental group. Many of these results also apply to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with nonempty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank-two fundamental group and all arithmetic 3-manifolds

Article information

Source
Duke Math. J., Volume 136, Number 1 (2007), 181-203.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1165244883

Digital Object Identifier
doi:10.1215/S0012-7094-07-13616-0

Mathematical Reviews number (MathSciNet)
MR2271299

Zentralblatt MATH identifier
1109.57015

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 30F40: Kleinian groups [See also 20H10] 20E07: Subgroup theorems; subgroup growth

Citation

Lackenby, Marc. Covering spaces of $3$ -orbifolds. Duke Math. J. 136 (2007), no. 1, 181--203. doi:10.1215/S0012-7094-07-13616-0. https://projecteuclid.org/euclid.dmj/1165244883


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References

  • M. Boileau, B. Leeb, and J. Porti, Geometrization of,$3$-dimensional orbifolds, Ann. of Math. (2) 162 (2005), 195--290.
  • J. Button, Strong Tits alternatives for compact $3$-manifolds with boundary, J. Pure Appl. Algebra 191 (2004), 89--98.
  • D. Cooper, C. D. Hodgson, and S. P. Kerckhoff, Three-Dimensional Orbifolds and Cone-Manifolds, MSJ Mem. 5, Math. Soc. Japan, Tokyo, 2000.
  • D. Cooper, D. D. Long, and A. W. Reid, Essential closed surfaces in bounded $3$-manifolds, J. Amer. Math. Soc. 10 (1997), 553--563.
  • D. Gabai, G. R. Meyerhoff, and N. Thurston, Homotopy hyperbolic $3$-manifolds are hyperbolic, Ann. of Math. (2) 157 (2003), 335--431.
  • M. Lackenby, Heegaard splittings, the virtually Haken conjecture and property $(\tau),$ Invent. Math. 164 (2006), 317--359.
  • —, Large groups, property $(\tau)$ and the homology growth of subgroups, preprint.
  • —, Some $3$-manifolds and $3$-orbifolds with large fundamental group, preprint.
  • M. Lackenby, D. D. Long, and A. W. Reid, Covering spaces of arithmetic $3$-orbifolds, preprint.
  • A. Lubotzky, Group presentations, $p$-adic analytic groups and lattices in $\rm SL\sb2(C)$ Ann. of Math. (2) 118 (1983), 115--130.
  • —, ``Dimension function for discrete groups'' in Proceedings of,Groups (St. Andrews, Scotland, 1985), London Math. Soc. Lecture Note Ser. 121, Cambridge Univ. Press, Cambridge, 1986, 254--262.
  • —, Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math. 125, Birkhäuser, Basel, 1994.
  • —, Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem, Ann. of Math. (2) 144 (1996), 441--452.
  • —, private communication, 2005.
  • A. Lubotzky and A. Mann, Powerful $p$-groups, II: $p$-adic analytic groups, J. Algebra 105 (1987), 506--515.
  • A. Lubotzky and D. Segal, Subgroup Growth, Progr. Math. 212, Birkhäuser, Basel, 2003.
  • J. G. Ratcliffe, ``Euler characteristics of $3$-manifold groups and discrete subgroups of $\rm SL(2, \Bbb C)$'' in Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Ill., 1985), J. Pure Appl. Algebra 44, Elsevier, Amsterdam, 1987, 303--314.
  • P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), 401--487.
  • W. P. Thurston, The geometry and topology of $3$-manifolds, lecture notes, Princeton Univ., Princeton, 1978.