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Quasiconformal homeomorphisms and dynamics. II: Structural stability implies hyperbolicity for Kleinian groups. (English) Zbl 0606.30044
This continues the author’s important series of papers of which I was [Ann. Math., II. Ser. 122, 401-418 (1985; Zbl 0589.30022)]. Let $$\Gamma_ 0\subset PS\ell (2,C)$$ be a finitely generated non-solvable subgroup. $$\Gamma_ 0$$ is structurally stable if all sufficiently near representations into PS$$\ell (2,C)$$ are injective. $$\Gamma_ 0$$ is non- rigid if there are arbitrarily close representations which are non- conjugate in PS$$\ell (2,C)$$. Theorem A states that a torsion-free group $$\Gamma_ 0$$ is structurally stable only if it is either rigid or is a discrete geometrically finite group with no non-trivial parabolic elements. In the latter case the action of $$\Gamma_ 0$$ is expanding on the limit set of $$\Gamma_ 0$$. The proof of Theorem A also shows that a neighbourhood of a non-rigid structurally stable $$\Gamma_ 0$$ in the variety of representations up to conjugation is nonsingular, has dimension 3 and consists of quasiconformal conjugates of $$\Gamma_ 0$$. The geometrically finite torsion free Kleinian groups turn out to be precisely the quasiconformally structurally stable groups in the sense of Bers.
Reviewer: I.N.Baker

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
structurally stable
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