Ending lamination theorem

In <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>, the ending lamination theorem, originally conjectured by <a href="/facts/William_Thurston/K1Qe8801">William Thurston</a> (1982) as the eleventh problem out of his <a href="/facts/Thurston%2527s_24_questions/Nl0upss5">twenty-four questions</a>, states that <a href="/facts/Hyperbolic_3-manifold/U8AAYhsw">hyperbolic 3-manifolds</a> with <a href="/facts/Finitely-generated_group/fVsFHrmb">finitely generated</a> <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groups</a> are determined by their topology together with certain "end invariants", which are geodesic <a href="/facts/Lamination_(topology)/NC7FJ8Kr">laminations</a> on some surfaces in the boundary of the manifold.
The ending lamination theorem is a generalization of the <a href="/facts/Mostow_rigidity_theorem/KXtuIhnr">Mostow rigidity theorem</a> to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces.
Minsky (2010) and Brock, Canary & Minsky (2012) proved the ending lamination conjecture for <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian</a> <a href="/facts/Surface_group/B2PbCFwC">surface groups</a>. In view of the <a href="/facts/Tameness_theorem/TsrXGtBT">Tameness theorem</a> this implies the ending lamination conjecture for all finitely generated Kleinian groups, from which the general case of ELT follows.

Ending lamination theorem open-in-new

Ending lamination theorem