The classification of punctured-torus groups

• 成果类型: Article
• 署名作者: Minsky, YN
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/120976
• 发表日期: 1999
• 页码: 559-626
• 关键词: generated kleinian-groups; mapping class group; hyperbolic 3-manifolds; teichmuller-spaces; orientable surface; short geodesics; BOUNDARIES; MANIFOLDS; homeomorphisms; CONVERGENCE

摘要

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.