The big three
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1, The Tameness Conjecture: The interior of every hyperbolic manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold.
2. The Density Conjecture: Every finitely generated Kleinian group G is the algebraic limit of geometrically finite groups isomorphic to G, Every group on the boundary of T(G) (The deformation space/Teichmuller space of G) is the algebraic limit of groups lying within T(G).
3. The Ending Lamination Conjecture: Hyperbolic manifolds M with finitely generated fundamental group are completely determined by the conformal structure of the boundary of M and the ending laminations of the geometrically infinite ends of M.
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