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Rigidity and topological conjugates of topologically tame Kleinian groups. (English) Zbl 0936.30031
A finitely-generated torsion-free Kleinian group \(\Gamma\) is topologically tame if \(H^3/\Gamma\) is homeomorphic to the interior of a compact 3-manifold. The main results here are quasiconformal rigidity theorems for such groups. More precisely the author proves that, if a homeomorphism \(h:H^3/ \Gamma_1\to H^3/ \Gamma_2\) preserves cusps, geometrically finite ends and ending laminations of geometrically infinite ends and an additional condition (IR), then \(\Gamma_1,\Gamma_2\) are \(q-c\) conjugate. The condition (IR) is that the injectivity radius at all points of both manifolds is bounded below. If \(\Gamma\) is freely indecomposable as a group, combining results of Bonahon and Thurston, shows that \(\Gamma\) is topologically tame. For freely indecomposable groups \(\Gamma_1,\Gamma_2\) also satisfying (IR) the rigidity theorem above had been proved by Y. N. Minsky [J. Am. Math. Soc. 7, No. 3, 539-588 (1994; Zbl 0808.30027)]. Thus the extension in this paper is to topologically tame groups which may be freely decomposable. Essential use is made of a branched-covering technique of Canary to extend Bonahon’s theory to freely decomposable groups. A particular application of this shows that, if two topologically tame Kleinian groups, which are not free and satisfy (IR) are topologically conjugate then they are \(q-c\) conjugate.

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