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Constructing geometrically infinite groups on boundaries of deformation spaces. (English) Zbl 1195.57040

This is a very interesting paper in the areas associated with the Ahlfors conjecture, the tameness conjecture and the Bers-Thurston density conjecture.
Recall that the tameness conjecture (recently proved by Minsky, Brock, Canary and Mazur) asserts that a topologically tame Kleinian group can be completely classified, up to conjugacy, by the four invariants: the homeomorphism type of the associated hyperbolic \(3\)-manifold; the parabolic loci; the conformal structure at infinity associated to the geometrically infinite ends; the ending laminations associated to the geometrically infinite ends.
The main result of the paper is to show that if the parabolic locus is empty, then any triple of the three then remaining invariants can be realised by a topologically tame Kleinian group without parabolic elements on the boundary of quasi-conformal deformation space as its homeomorphism type.
It seems worth to remark that the paper represents the basis for a more recent paper of the author, in which he gives a confirmation of the Bers-Thurston density conjecture, which asserts that every finitely generated Kleinian group can be obtained as an algebraic limit of quasi-conformal deformations of a minimally parabolic geometrically finite Kleinian group.

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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References:

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