Finite Sheeted Covers and the Virtual Haken Conjecture for Hyperbolic 3-Manifolds

This monograph presents a rigorous exposition of the resolution of the Virtual Haken Conjecture for closed hyperbolic 3-manifolds, a landmark achievement in low-dimensional topology. We systematically reconstruct the pathway from Thurston's Geometrization Conjecture to Agol's proof of the Virtual Haken Conjecture, elucidating the critical roles of the Kahn-Markovic Surface Subgroup Theorem and Wise's theory of special cube complexes. We detail the construction of immersed 1-injective quasi-Fuchsian surfaces, the subsequent cubulation of the fundamental group, and the final deduction of virtual specialness. Furthermore, we explore the implications of the Virtual Fibering Conjecture, demonstrating that every closed hyperbolic 3-manifold admits a finite sheeted cover that fibers over the circle. The interplay between geometric group theory, particularly the study of Right-Angled Artin Groups (RAAGs), and hyperbolic geometry is emphasized.