Tameness theorem

Remarks

The statement was conjectured by Marden and is also called the Marden conjecture. It was proved in 2004 by Agol and, independently, by Calegari and Gabai. Both proofs approximate the ends of $M$ by surfaces of controlled geometry, building on work of Canary and others on tame ends. For example, if $M = \mathbb{H}^3 / \Gamma$ with $\Gamma$ a quasi-Fuchsian group isomorphic to the fundamental group of a closed surface $S$, then $M$ is homeomorphic to $S \times \R$, the interior of $S \times [0, 1]$. The hypothesis that $\pi_1(M)$ is finitely generated is needed: there are hyperbolic $3$-manifolds with infinitely generated fundamental group that are not the interior of any compact manifold. Among its consequences is the Ahlfors measure conjecture: the limit set of a finitely generated Kleinian group is either the whole sphere or has Lebesgue measure zero.

Going deeper

For a student

Tameness is the topological input for the classification of hyperbolic $3$-manifolds with finitely generated fundamental group. Together with the ending lamination theorem and the density theorem, it gives a complete picture of the deformation space of Kleinian groups. It also implies that such a manifold has finitely many ends, each homeomorphic to a surface times a half-line.