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 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
or (without figures) as a
It is also available through the
LANL XXX Math Archives.
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