## Institute for Mathematical Sciences

## Preprint ims97-6

** Y. Minsky**
* The Classification of Punctured-Torus Groups*

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.
(revised version of May 1988)

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