2013/09/30 by Yi Liu, Vladimir Marković, Vladimir Markovic
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homology (biology) #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Injective function #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Submanifold #math.GT #msc:20H10 #msc:57M05
paper · pdf · doi:10.1215/00127094-3167744
published as Duke Math. J. 164, no. 14 (2015), 2723-2808 · 66 pages, 8 figures, with additional explanations and minor corrections
arxiv created 2014/10/29 · openalex publication_date 2015/10/26 · arxiv updated 2015/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Among other things, we prove the following two topological statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positive integral multiple represented by an oriented connected closed π1-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, π1-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented connected compact π1-injectively immersed quasi-Fuchsian subsurface. In, we exploit techniques developed by Kahn and Markovic but we only distill geometric and topological ingredients from those papers, so no hard analysis is involved in this article.