Tameness of Hyperbolic 3-manifolds

Marden conjectured that a hyperbolic 3-manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3-manifold topologists. We prove this conjecture in theorem 10.2, actually in slightly more generality for PNC manifolds with hyperbolic cusps (the cusped case is reduced to the non-cusped case in section 6, see the next section for definitions of this terminology). Many special cases of Marden’s conjecture have been resolved and various criteria for tameness have been developed, see [42, 55, 9, 19, 21, 27, 54, 12, 13, 38]. Bonahon resolved the case where π1(M) is indecomposable, that is π1(M) 6= A ∗ B. This was generalized by Canary [17, 18] to the case of PNC manifolds with indecomposable fundamental group. Various other cases of limits of tame manifolds being tame have been resolved, culminating in the proof by Brock and Souto that algebraic limits of tame manifolds are tame [55, 21, 12, 13]. In a certain sense, Canary’s covering theorem provides other examples of tame manifolds [20], whereby tameness of a cover implies tameness of a quotient under certain circumstances. Conversely, Canary and Thurston showed that covers (with finitely generated fundamental group) of tame hyperbolic manifolds of infinite volume are tame (this is a purely topological result) [19]. Bonahon in fact proved that hyperbolic 3-manifolds with indecomposable fundamental group are geometrically tame [9]. Geometric tameness was a condition formulated by Thurston (and proved in some special cases by him) which implies that the ends are either geometrically finite or simply degenerate [55]. Canary showed that tame manifolds are geometrically tame [18]. He also showed (generalizing an argument of Thurston [55]) that geometric tameness implies the Ahlfors measure conjecture [3]. Canary’s arguments were generalized for PNC manifolds by Yong Hou [34]. In fact, these arguments give a geometric proof of the Ahlfors finiteness theorem [3]. Other corollaries for Kleinian groups are noted in [19]. Assuming the geometric tameness conjecture, Thurston conjectured that a hyperbolic 3-manifold N is determined by its end invariants: the topological type of N , the conformal structure of the domain of discontinuity of each geometrically finite end, and the ending lamination of each simply degenerate end. This was resolved