Let G be a torsion-free hyperbolic group and let n  6 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n − 1)-dimensional sphere.

Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

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