Quasiconformal homeomorphisms and dynamics. II: Structural stability implies hyperbolicity for Kleinian groups.

*(English)*Zbl 0606.30044This 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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