Homology of curves and surfaces in closed hyperbolic 3-manifolds

Abstract

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed ${\pi }_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, ${\pi }_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact ${\pi }_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques developed earlier by Kahn and Markovic about good pants constructions, but we only distill geometric and topological ingredients from their papers so no hard analysis is involved in this paper.

66 pages, 8 figures, with additional explanations and minor corrections

Cited by in corpus (8)

• Virtual Domination of $3$-manifolds
• Virtual 1-domination of 3-manifolds
• Positive simplicial volume implies virtually positive Seifert volume for $3$-manifolds
• Incompressible surfaces in rank one locally symmetric spaces
• Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups
• Virtual Domination of 3-manifolds III
• The panted cobordism group of cusped hyperbolic 3-manifolds
• Virtual domination of 3-manifolds II