Geometry & Topology

Surface subgroups and handlebody attachment

Vivien R Easson

Full-text: Open access


The main theorem of this paper generalizes recent results in Dehn surgery to the case of handlebody attachment. We consider attaching handlebodies and solid tori to the boundary of an irreducible, boundary-irreducible, atoroidal and acylindrical 3–manifold. We show that for a large class of homeomorphisms attaching these handlebodies, the fundamental group of the resulting manifold contains the fundamental group of a closed surface of genus at least two.

Article information

Geom. Topol., Volume 10, Number 1 (2006), 557-591.

Received: 4 March 2004
Revised: 19 July 2005
Accepted: 25 March 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57N35: Embeddings and immersions 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

3-manifold handlebody attachment surface subgroup


Easson, Vivien R. Surface subgroups and handlebody attachment. Geom. Topol. 10 (2006), no. 1, 557--591. doi:10.2140/gt.2006.10.557.

Export citation


  • A Abrams, S Schleimer, Distances of Heegaard splittings, Geom. Topol. 9 (2005) 95–119
  • A Bart, Surface groups in some surgered manifolds, Topology 40 (2001) 197–211
  • A Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994) 207–225
  • F Bonahon, }, Ann. of Math. $(2)$ 124 (1986) 71–158
  • F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139–162
  • A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press, Cambridge (1988)
  • D Cooper, D D Long, Some surface subgroups survive surgery, Geom. Topol. 5 (2001) 347–367
  • D Cooper, D D Long, A W Reid, Essential closed surfaces in bounded $3$-manifolds, J. Amer. Math. Soc. 10 (1997) 553–563
  • S Gallot, D Hulin, J Lafontaine, Riemannian geometry, Universitext, Springer, Berlin (2004)
  • J Hass, J H Rubinstein, S Wang, Boundary slopes of immersed surfaces in 3-manifolds, J. Differential Geom. 52 (1999) 303–325
  • K Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics 761, Springer, Berlin (1979)
  • M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser, Boston (2001)
  • S P Kerckhoff, The measure of the limit set of the handlebody group, Topology 29 (1990) 27–40
  • T Kobayashi, Heights of simple loops and pseudo-Anosov homeomorphisms, from: “Braids (Santa Cruz, CA, 1986)”, Contemp. Math. 78, Amer. Math. Soc., Providence, RI (1988) 327–338
  • M Lackenby, Attaching handlebodies to 3-manifolds, Geom. Topol. 6 (2002) 889–904
  • T Li, Immersed essential surfaces in hyperbolic 3-manifolds, Comm. Anal. Geom. 10 (2002) 275–290
  • H Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems 6 (1986) 99–116
  • K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (1998)
  • W Menasco, A W Reid, Totally geodesic surfaces in hyperbolic link complements, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 215–226
  • J W Morgan, H Bass, The Smith conjecture, Pure and Applied Mathematics 112, Academic Press, Orlando, FL (1984)
  • W P Thurston, The geometry and topology of 3–manifolds, Princeton University Lecture Notes (1982)