Among other things, we prove the following two topological statements about closed hyperbolic -manifolds. First, every rational second homology class of a closed hyperbolic -manifold has a positive integral multiple represented by an oriented connected closed -injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, -injectively immersed oriented closed -submanifold in a closed hyperbolic -manifold has an equidegree finite cover which bounds an oriented connected compact -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.
"Homology of curves and surfaces in closed hyperbolic -manifolds." Duke Math. J. 164 (14) 2723 - 2808, 1 November 2015. https://doi.org/10.1215/00127094-3167744