Algebraic & Geometric Topology

Counting immersed surfaces in hyperbolic 3–manifolds

Joseph D Masters

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Abstract

We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3–manifold groups. For any closed hyperbolic 3–manifold, we show that there is an upper bound on this number which grows factorially with g. We also give a class of closed hyperbolic 3–manifolds for which there is a lower bound of the same type.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 835-864.

Dates
Received: 20 October 2004
Accepted: 13 June 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796433

Digital Object Identifier
doi:10.2140/agt.2005.5.835

Mathematical Reviews number (MathSciNet)
MR2153105

Zentralblatt MATH identifier
1082.57013

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57N16: Geometric structures on manifolds [See also 57M50] 57M27: Invariants of knots and 3-manifolds

Keywords
surface subgroups bending pleated surfaces reflection orbifolds

Citation

Masters, Joseph D. Counting immersed surfaces in hyperbolic 3–manifolds. Algebr. Geom. Topol. 5 (2005), no. 2, 835--864. doi:10.2140/agt.2005.5.835. https://projecteuclid.org/euclid.agt/1513796433


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References

  • B H Bowditch, G Mess, A $4$-dimensional Kleinian group, Trans. Amer. Math. Soc. 344 (1994) 391–405\relax
  • J Elstrodt, F Grunewald, J Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin (1998)\relax
  • D G James, C Maclachlan, Fuchsian subgroups of Bianchi groups, Trans. Amer. Math. Soc. 348 (1996) 1989–2002\relax
  • Alexander Lubotzky, Counting finite index subgroups, from: “Groups '93 Galway/St. Andrews, Vol. 2”, London Math. Soc. Lecture Note Ser. 212, Cambridge Univ. Press, Cambridge (1995) 368–404\relax
  • C Maclachlan, A W Reid, Parametrizing Fuchsian subgroups of the Bianchi groups, Canad. J. Math. 43 (1991) 158–181\relax
  • Teruhiko Soma, Virtual fibers in hyperbolic $3$-manifolds, Topology Appl. 41 (1991) 179–192\relax
  • Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978) 555–565\relax
  • William Thurston, The Geometry and Topology of Three-Manifolds, lecture notes, Princeton University (1978-1980) http://www.msri.org/publications/books/gt3m/