The Classification of Punctured-Torus Groups

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...

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Bibliographic Details
Published in:Annals of mathematics 1999-03, Vol.149 (2), p.559-626
Main Author: Minsky, Yair N.
Format: Article
Language:English
Subjects:
Homeomorphism
Laminates
Mathematical cusps
Mathematical theorems
Perceptron convergence procedure
Topological theorems
Triangulation
Vertices
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Summary: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.
ISSN:0003-486X
DOI:10.2307/120976