1998/07/31 by Yair N. Minsky · 1 citation
Mathematics · #math.GT #math.DS #msc:30F40 #msc:57M50
published as Ann. of Math. (2) 149 (1999), no. 2, 559-626 · 67 pages, published version
arxiv created 1999/03/01 · arxiv updated 2009/11/30
Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers' conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.