## Geometry & Topology

### Surface subgroups and handlebody attachment

Vivien R Easson

#### Abstract

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

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

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

https://projecteuclid.org/euclid.gt/1513799715

Digital Object Identifier
doi:10.2140/gt.2006.10.557

Mathematical Reviews number (MathSciNet)
MR2224465

Zentralblatt MATH identifier
1128.57019

#### Citation

Easson, Vivien R. Surface subgroups and handlebody attachment. Geom. Topol. 10 (2006), no. 1, 557--591. doi:10.2140/gt.2006.10.557. https://projecteuclid.org/euclid.gt/1513799715

#### References

• 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)