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Annals of Mathematics, II. Series Vol. 149, No. 2, pp. 559626 (1999) 

The classification of puncturedtorus groupsYair N. MinskyReview from Zentralblatt MATH: The general classification problem for Kleinian groups remains open, even though a conjectural picture of the solution has been in place since the late 70's. This paper verifies the picture for puncturedtorus groups, which is the simplest class of Kleinian groups with nontrivial deformation theory. A puncturedtorus group is a discrete faithful representation of the fundamental group of the oncepunctured torus into $\text{PSL}_2(\Bbb C)$ such that the boundary loop is mapped into a parabolic element. To each such group one can associate an ordered pair of socalled end invariants lying in the complement of the diagonal of $D\times D$ where $D$ is a closed $2$disk. The interior of $D$ is identified with the Teichmüller space and the boundary corresponds to the space of measured laminations on the punctured torus. The main theorem says that the map which associates the end invariants to the conjugacy class of a puncturedtorus group is a bijection with continuous inverse. (Surprisingly, the map itself is discontinuous.) In particular, this solves Thurston's ending lamination conjecture for puncturedtorus groups. Furthermore, it is proved that each Bers slice is a closed disk, and each Maskit slice is a closed disk with one boundary point removed. The main step of the proof is to get quasiisometric control of the group in terms of continuousfraction expansions of the end invariants. One application is a proof of Bers's conjecture (for puncturedtorus groups) which says that all degenerate groups in the Bers slice are limits of quasiFuchsian groups. Another application is the quasiconformal rigidity theorem: two puncturedtorus groups on the Riemann sphere are topologically conjugate iff they are quasiconformally conjugate. Reviewed by Igor Belegradek Keywords: punctured torus; ending lamination Classification (MSC2000): 30F40 30F60 57M50 32G15 Full text of the article:
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