The classification of punctured-torus groups

Yair N. Minsky

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arXiv · math/9807001

The classification of punctured-torus groups

Yair N. Minsky

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Abstract

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.

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Yair N. Minsky. 1999-03-01. The classification of punctured-torus groups. https://arxiv.org/abs/math/9807001

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