The classification of punctured-torus groups.

*(English)*Zbl 0939.30034The 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 punctured-torus groups, which is the simplest class of Kleinian groups with nontrivial deformation theory.

A punctured-torus group is a discrete faithful representation of the fundamental group of the once-punctured torus into \(\text{PSL}_2(\mathbb C)\) such that the boundary loop is mapped into a parabolic element. To each such group one can associate an ordered pair of so-called 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 punctured-torus group is a bijection with continuous inverse. (Surprisingly, the map itself is discontinuous.) In particular, this solves Thurston’s ending lamination conjecture for punctured-torus 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 quasi-isometric control of the group in terms of continuous-fraction expansions of the end invariants.

One application is a proof of Bers’s conjecture (for punctured-torus groups) which says that all degenerate groups in the Bers slice are limits of quasi-Fuchsian groups. Another application is the quasiconformal rigidity theorem: two punctured-torus groups on the Riemann sphere are topologically conjugate iff they are quasiconformally conjugate.

A punctured-torus group is a discrete faithful representation of the fundamental group of the once-punctured torus into \(\text{PSL}_2(\mathbb C)\) such that the boundary loop is mapped into a parabolic element. To each such group one can associate an ordered pair of so-called 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 punctured-torus group is a bijection with continuous inverse. (Surprisingly, the map itself is discontinuous.) In particular, this solves Thurston’s ending lamination conjecture for punctured-torus 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 quasi-isometric control of the group in terms of continuous-fraction expansions of the end invariants.

One application is a proof of Bers’s conjecture (for punctured-torus groups) which says that all degenerate groups in the Bers slice are limits of quasi-Fuchsian groups. Another application is the quasiconformal rigidity theorem: two punctured-torus groups on the Riemann sphere are topologically conjugate iff they are quasiconformally conjugate.

Reviewer: Igor Belegradek (Hamilton/Ontario)