## Algebraic & Geometric Topology

### Finite-volume hyperbolic $3$–manifolds contain immersed quasi-Fuchsian surfaces

#### Abstract

The paper contains a new proof that a complete, non-compact hyperbolic $3$–manifold with finite volume contains an immersed, closed, quasi-Fuchsian surface.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 2 (2015), 1199-1228.

Dates
Revised: 21 August 2014
Accepted: 26 August 2014
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895803

Digital Object Identifier
doi:10.2140/agt.2015.15.1199

Mathematical Reviews number (MathSciNet)
MR3342690

Zentralblatt MATH identifier
06442394

#### Citation

Baker, Mark D; Cooper, Daryl. Finite-volume hyperbolic $3$–manifolds contain immersed quasi-Fuchsian surfaces. Algebr. Geom. Topol. 15 (2015), no. 2, 1199--1228. doi:10.2140/agt.2015.15.1199. https://projecteuclid.org/euclid.agt/1511895803

#### References

• J,W Anderson, Intersections of topologically tame subgroups of Kleinian groups, J. Anal. Math. 65 (1995) 77–94
• M,D Baker, D Cooper, A combination theorem for convex hyperbolic manifolds, with applications to surfaces in $3$–manifolds, J. Topol. 1 (2008) 603–642
• M,D Baker, D Cooper, Conservative subgroup separability for surfaces with boundary, Algebr. Geom. Topol. 13 (2013) 115–125
• B,H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245–317
• 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
• M Culler, P,B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. 117 (1983) 109–146
• M Culler, P,B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537–545
• J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012) 1127–1190
• A Kurosch, Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934) 647–660
• J,D Masters, X Zhang, Closed quasi-Fuchsian surfaces in hyperbolic knot complements, Geom. Topol. 12 (2008) 2095–2171
• J,D Masters, X Zhang, Quasi-Fuchsian surfaces in hyperbolic link complements (2009)
• T Soma, Function groups in Kleinian groups, Math. Ann. 292 (1992) 181–190
• W,P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357–381