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Finite-sheeted covering spaces of 3-manifolds, engulfing, and solenoids

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Will Cavendish, Princeton

Abstract:
The study of finite-sheeted covering spaces of 3-manifolds has been invigorated in recent years by work of Agol, Kahn-Markovich and Wise. This work makes significant strides towards the resolution of Thurston's virtually fibered conjecture, which states that every hyperbolic 3-manifold has a finite-sheeted covering space homeomorphic to a surface bundle over the circle. Given recently announced work of Wise, the virtually fibered conjecture holds for Haken hyperbolic 3-manifolds, which are those hyperbolic 3-manifolds that contain embedded \pi_1-injective surfaces. A positive resolution of the virtually Haken conjecture, which states that every aspherical 3-manifold with infinite fundamental group has a covering space containing such a surface, would therefore imply that the virtually fibered conjecture holds for all hyperbolic 3-manifolds. I will begin this talk by giving an introductory survey of the theory of finite sheeted covering spaces of 3-manifolds highlighting recent developments in the subject. I will then discuss the concept of "engulfing." A subgroup H of a group G is said to be "engulfed" if H is contained in a finite index subgroup of G other than G itself. An argument due to Shalen, together with the resolution of the surface subgroup conjecture for hyperbolic 3-manifolds by Kahn and Markovic, shows that if surface subgroups of hyperbolic 3-manifold groups are engulfed, then hyperbolic 3-manifolds are virtually Haken. I will present this argument, and then conclude by discussing some reformulations of the engulfing condition in terms of objects called solenoids, which are generalizations of finite sheeted covering spaces given by taking inverse limits of towers of finite-sheeted covers.