Deformations of hyperbolic structures.

*(English)*Zbl 0577.53041Let G be a discrete group of isometries of the hyperbolic space \(H^ n\). There is the natural inclusion \(H^ n/G\subset H^{n+1}/G\) for which \(H^ n/G\) is totally geodesic. In the paper bending deformations of the hyperbolic structure on the orbifold \(H^ n/G\) (on n-manifold if G is torsion-free) are studied induced by quasiconformal deformations of the \((n+1)\)-dimensional hyperbolic orbifold \(H^{n+1}/G\). Since G acts on the conformal sphere \(S^ n=\partial H^{n+1}\) as a discrete group of MĂ¶bius transformations which leave some ball invariant, such deformations correspond to the space of quasi-Fuchsian groups which are quasiconformally conjugated on \(S^ n\) to the group G.

For the first time the non-triviality of such spaces (for rigid groups \(G\subset Isom H^ n\), \(n\geq 3)\) was established in the papers of the reviewer and A. V. Tjetjenov [Dokl. Akad. Nauk SSSR 239, No.1, 14- 17 (1978; Zbl 0412.20040)] and the reviewer [Ann. Math. Stud. 97, 21-31 (1981; Zbl 0464.30037)]. The matter was greatly clarified by Thurston’s ”Mickey Mouse” example [cf. W. Thurston, Geometry and topology of 3-manifolds (mimeographed notes) (1978/79), and D. Sullivan, Bull. Am. Math. Soc., New. Ser. 6, 57-73 (1982; Zbl 0489.58027)] who showed that in this case there arise bendings of \(H^ n/G\) along some totally geodesic submanifold.

Developing this idea the author constructed bending deformations for a large class of groups \(G\subset Isom H^ n\) (by means of a limit of some sequence of simple bendings of \(H^{n+1})\). The main result of the paper is the construction of smooth embedding of an m-dimensional ball into the space \(Hom(G, Isom H^{n+1})/Isom H^{n+1}\) induced by bending deformations.

Independent approaches to bending deformations are contained in the papers: the reviewer, Thurston’s bends and geometric deformations of conformal structures, Proc. Complex Anal. Appl., Varna, May 1985 (to appear); D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Proc. Conf., in honour of G. D. Mostow, Yale Univ. (to appear). In the latter paper the existence of singularities of the spaces \(Hom(G, Isom H^{n+1})\) and \(Hom(G, Isom H^{n+1})/Isom H^{n+1}\) is proved.

For the first time the non-triviality of such spaces (for rigid groups \(G\subset Isom H^ n\), \(n\geq 3)\) was established in the papers of the reviewer and A. V. Tjetjenov [Dokl. Akad. Nauk SSSR 239, No.1, 14- 17 (1978; Zbl 0412.20040)] and the reviewer [Ann. Math. Stud. 97, 21-31 (1981; Zbl 0464.30037)]. The matter was greatly clarified by Thurston’s ”Mickey Mouse” example [cf. W. Thurston, Geometry and topology of 3-manifolds (mimeographed notes) (1978/79), and D. Sullivan, Bull. Am. Math. Soc., New. Ser. 6, 57-73 (1982; Zbl 0489.58027)] who showed that in this case there arise bendings of \(H^ n/G\) along some totally geodesic submanifold.

Developing this idea the author constructed bending deformations for a large class of groups \(G\subset Isom H^ n\) (by means of a limit of some sequence of simple bendings of \(H^{n+1})\). The main result of the paper is the construction of smooth embedding of an m-dimensional ball into the space \(Hom(G, Isom H^{n+1})/Isom H^{n+1}\) induced by bending deformations.

Independent approaches to bending deformations are contained in the papers: the reviewer, Thurston’s bends and geometric deformations of conformal structures, Proc. Complex Anal. Appl., Varna, May 1985 (to appear); D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Proc. Conf., in honour of G. D. Mostow, Yale Univ. (to appear). In the latter paper the existence of singularities of the spaces \(Hom(G, Isom H^{n+1})\) and \(Hom(G, Isom H^{n+1})/Isom H^{n+1}\) is proved.

Reviewer: B.N.Apanasov

##### MSC:

53C40 | Global submanifolds |

57S25 | Groups acting on specific manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

##### Keywords:

Kleinian groups; hyperbolic space; totally geodesic; bending deformations; quasiconformal deformations; quasi-Fuchsian groups
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\textit{C. Kourouniotis}, Math. Proc. Camb. Philos. Soc. 98, 247--261 (1985; Zbl 0577.53041)

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