Tameness theorem

In mathematics, the Tame PRIME guess is a going back to Albert Marden conjecture in the theory of Kleinian groups in 3-dimensional topology, which was proved by Ian Agol, Danny Calegari and David Gabai 2004.

Statement

Each complete 3-dimensional hyperbolic manifold with finitely generated fundamental group is topologically tame, ie is homeomorphic to the interior of a compact manifold.

Ends of hyperbolic 3-manifolds

That every orientable 3-dimensional hyperbolic manifold complete can be disassembled with finitely generated fundamental group into a compact core ( which is homeomorphic to ) and a finite number of contiguous "ends" which are of the form from the topological tameness follows immediately. The surfaces are homeomorphic to the connected components of.

Role of hyperbolicity

The assumption that is hyperbolic, plays an essential role in the proof of Tame PRIME guess. There are counter-examples of ( non- hyperbolic ) 3-manifolds with finitely generated fundamental group, whose ends are not tame.