The Classification of Punctured-Torus Groups
(Final Revision)

May, 1998


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.

The original draft of this paper, which appeared here in September of 1995, has been considerably revised and (hopefully) improved. The final version is available as a Postscript file, or (without figures) as a DVI file. It is also available through the LANL XXX Math Archives.

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