Geometry & Topology

Group negative curvature for 3–manifolds with genuine laminations

David Gabai and William H Kazez

Full-text: Open access

Abstract

We show that if a closed atoroidal 3–manifold M contains a genuine lamination, then it is group negatively curved in the sense of Gromov. Specifically, we exploit the structure of the non-product complementary regions of the genuine lamination and then apply the first author’s Ubiquity Theorem to show that M satisfies a linear isoperimetric inequality.

Article information

Source
Geom. Topol., Volume 2, Number 1 (1998), 65-77.

Dates
Received: 5 August 1997
Revised: 9 May 1998
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882902

Digital Object Identifier
doi:10.2140/gt.1998.2.65

Mathematical Reviews number (MathSciNet)
MR1619168

Zentralblatt MATH identifier
0905.57011

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R30: Foliations; geometric theory 57M07: Topological methods in group theory 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20F32 57M30: Wild knots and surfaces, etc., wild embeddings

Keywords
lamination essential lamination genuine lamination group negatively curved word hyperbolic

Citation

Gabai, David; Kazez, William H. Group negative curvature for 3–manifolds with genuine laminations. Geom. Topol. 2 (1998), no. 1, 65--77. doi:10.2140/gt.1998.2.65. https://projecteuclid.org/euclid.gt/1513882902


Export citation

References

  • M Bestvina, M Feighn, A Combination Theorem for Negatively Curved Groups, J. Diff. Geom. 35 (1992) 85–101
  • M Brittenham, Essential Laminations in Seifert Fibred Spaces, Topology 32 (1993) 61–85
  • M Brittenham, Essential Laminations and Haken Normal Form I, Pacific J. Math. 168 (1995) 217–234
  • E Claus, Thesis, U Texas, Austin(1990)
  • C Delman, R Roberts (in preparation)
  • D Gabai, Problems in the Geometric Theory of Foliations and Laminations on 3–Manifolds, Geometric Topology 2, edited W H Kazez, AMS/IP (1997) 1–33
  • D Gabai, Essential Laminations and Kneser Normal Form (in preparation)
  • D Gabai, \emph{The Ubiquitous Nature of Quasi-Minimal Semi-Euclidean Laminations in 3–Manifolds, Surveys in Differential Geometry 5, International Press
  • D Gabai, W H Kazez, Homotopy, Isotopy and Genuine Laminations of 3–Manifolds, Geometric Topology 1 edited W H Kazez, AMS/IP (1997) 123–138
  • D Gabai, W H Kazez, Finiteness of the Mapping Class Group for Atoroidal 3–Manifolds with Genuine Laminations, (preprint)
  • D Gabai, U Oertel, Essential Laminations in 3–Manifolds, Ann. Math. 130 (1989) 41–73
  • M Gromov, Hyperbolic Groups, MSRI Pubs. 8 75-2-64
  • W Haken, Theorie der Normal Flachen, Acta. Math. 105 (1961) 245–375
  • K Johannson, Homotopy Equivalences of 3–Manifolds with Boundary, Springer LNM 761 (1979)
  • W Jaco, P Shalen, Seifert Fibered Spaces in 3–Manifolds, Mem. AMS 21 (1979)
  • L Mosher, Combintorics of Pseudo-Anosov Flows, (in preparation)
  • R Naimi, Constructing essential laminations in $2$–bridge knot surgered $3$–manifolds, Pacific J. Math. 180 (1997) 153–186
  • W P Thurston, Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry, Proc. Symp. Pure Math. 39 (1979) 87–111
  • W P Thurston, \emph{On the Geometry and Dynamics of Diffeomorphisms of Surfaces Bull. Amer. Math.Soc. 19 (1988) 417–431
  • Y Q Wu, Dehn Surgery on Arborescent Knots, J. Diff. Geom. 43 (1996) 171–197