Algebraic & Geometric Topology

The co-rank conjecture for 3–manifold groups

Christopher J Leininger and Alan W Reid

Full-text: Open access


In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3–manifold group (also known as the cut number) is bounded below by one-third the first Betti number.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 37-50.

Received: 19 November 2001
Revised: 16 January 2002
Accepted: 27 January 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 57M50: Geometric structures on low-dimensional manifolds 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]

3–manifolds co-rank pseudo-Anosov


Leininger, Christopher J; Reid, Alan W. The co-rank conjecture for 3–manifold groups. Algebr. Geom. Topol. 2 (2002), no. 1, 37--50. doi:10.2140/agt.2002.2.37.

Export citation


  • J S Birman, R Craggs, The $\mu$-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold, Trans. Amer. Math. Soc. 237 (1978) 283–309
  • F Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup.(4) 16 (1983) 237–270
  • A J Casson, Notes from a lecture given at U.C. Santa Barbara, Dec. (1979)
  • A J Casson, C McA Gordon, Reducing Heegaard Splittings, Topology Appl. 27 (1987) 275–283
  • A J Casson, D D Long, Algorithmic compressions of surface automorphisms, Invent. Math. 81 (1985) 295–303
  • R E Gompf, A I Stipsicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics Series 20 American Mathematical Society, Providence, RI (1999)
  • J Hempel, Three-manifolds as viewed from the curve complex, Topology 40 (2001) 631–657
  • S L Harvey, On the cut number of a 3-manifold, Preprint
  • N V. Ivanov, Subgroups of Teichmüller Modular Groups, Translations of Math. Monographs 115 American Math. Society Publications (1992)
  • K Johannson, D Johnson, Non-bording diffeomorphisms of surfaces which act trivially on homology, Preprint
  • D Johnson, Quadratic forms and the Birman-Craggs homomorphisms, Trans. Amer. Math. Soc. 261 (1980) 235–254
  • D D Long, H R Morton, Hyperbolic 3-manifolds and surface automorphisms, Topology 25 (1986) 575–583
  • D D Long, A W Reid, Integral points on character varieties, Preprint
  • W Magnus, A Karrass, D Solitar, Combinatorial Group Theory, Dover Publications, Inc. (1976)
  • J-P Otal, Thurston's hyperbolization of Haken manifolds, Surveys in differential geometry Vol. III (Cambridge, MA, 1996) 77–194, Int. Press, Boston, MA (1998)
  • D Rolfsen, Knots and Links, Publish or Perish, Inc. (1990)
  • A Sikora, Cut numbers of 3-manifolds, Preprint
  • J Stallings, Geometric ideas about the corank of a group, Lecture given at the “Groups and Low-Dimensional Topology Conference”, CRM, Université de Montréal June 25-July13 (2001).
  • W P Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, Preprint
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc.(N.S.) 19 (1988) 417–431