Deformations of representations of discrete subgroups of \(SO(3,1)\). (English) Zbl 0828.57009

A hyperbolic 3-dimensional orbifold \(M\) has a holonomy representation \(\rho_0 : \pi_1 (M) \to \text{Isom} (H^3)\), whose conjugacy class is denoted by \([\rho_0]\). This can be regarded as a point in the deformation space \(R (\pi_1 (M),n)\) defined to be \(\operatorname{Hom} (\pi_1 (M), \text{Isom} (H^n))/ \text{Isom}_+ (H^n)\). When \([\rho_0]\) is an isolated point of \(R (\pi_1 (M), 4)\), \(\rho_0\) is called locally rigid.
The author’s general conjecture is that when \(M\) is closed, \(\rho_0\) is not locally rigid if and only if \(M\) contains an incompressible 2- suborbifold which is not a virtual fiber in a fiber bundle over \(S^1\). In this paper, various theorems and examples are given that support the conjecture. First, let \(\Phi\) be a compact convex finite-sided polyhedron in \(H^3\) such that each vertex of \(\Phi\) belongs to precisely 3 edges along which two faces meet, and let \(\Gamma_\Phi\) be the group generated by reflections in the faces of \(\Phi\). Assuming that \(\Gamma_\Phi\) is discrete and has \(\Phi\) as a fundamental polyhedron, the author proves that near the class of the inclusion imbedding \(\Gamma_\Phi \to \text{Isom} (H^3)\), the space \(R (\Gamma_\Phi, 4)\) is smooth and has dimension \(f - 4\), where \(f\) is the number of faces of \(\Phi\). In contrast, the author proves that if \(M_{(p,q)}\) is obtained by Dehn surgery on a fixed 2-bridge knot \(K\) in \(S^3\), then for infinitely many coprime pairs \((p,q)\), the groups \(\pi_1 (M_{(p,q)})\) are locally rigid in SO(4,1). Specific examples are also given of each of the following: (1) a reflection group having a deformation which is nontrivial on cusp structures, (2) a group \(G\) with a pair of bending deformations which define directions spanning a 2-dimensional plane tangent to \(R (G,4)\), and (3) a group \(H\) which does not admit bending deformations, but has \(R (H,4)\) 2-dimensional. The proofs of these theorems and the analysis of the examples use a variety of techniques from hyperbolic geometry, Lie theory, and algebraic geometry.


57M50 General geometric structures on low-dimensional manifolds
22E40 Discrete subgroups of Lie groups
57R30 Foliations in differential topology; geometric theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI EuDML


[1] Apanasov, B.: Bending and stamping deformations of hyperbolic maniforlds, Ann. Global. Anal. Geom.8(no. 1), 3-12 (1990) · Zbl 0705.53013
[2] Apanasov, B., Tetenov, A.: On existence of nontrivial quasiconformal deformations of Kleinian groups in space. Dokl. Sov. Acad. Sci.239, 14-17 (1978)
[3] Goldman, W.: Geometric structures on manifolds and varieties of representations. (Contemp. Math. vol. 74) Providence, RI: Am. Math. Soc. 1987
[4] Gromov, M., Lawson, H.B., Thurston, W.: Hyperbolic 4-manifolds and conformally flat 3-manifolds. Publ. Math., Inst. Hautes ?tud. Sci.68, 27-45 (1988) · Zbl 0692.57012
[5] Graland, R., Raghunathan, M.: Fundamental domains for lattices in rank 1 semisimple Lie groups. Ann. Math.92, 279-326 (1970) · Zbl 0206.03603
[6] Hatcher, A., Thurston, W.: Incompressible surfaces in 2-bridge knot complement. Invent. Math.79, F. 2, 225-247 (1985) · Zbl 0602.57002
[7] Johnson, D., Millson, J.: Deformation spaces associated to compact hyperbolic manifolds. In: Howe, R. (ed.) Discrete groups in geometry and analysis. Papers in honor of G. Mostow on his sixtieth birthday. (Prog. Math., vol. 67, pp. 48-106) Boston Basel Stuttgart: Birkh?user 1987
[8] Kapovich, M.: Flat conformal structures on 3-manifolds. Novosibirsk: Institute of Mathematics (Preprint 17, 1987) · Zbl 0631.53021
[9] Kapovich, M.: Flat conformal structures on 3-manifolds (survey). In: Bokut, L.A. et al. (eds.) Proceedings of International Conference on Algebra, Novosibirsk, 1989. (Contemp. Math. vol. 131.1, pp. 551-570) Providence, RI: Am. Math. Soc. 1992 · Zbl 0771.53022
[10] Kapovich, M.: Deformations of representations of discrete cocompact subgroups ofSO(3, 1). Inst. Hautes ?tud. Sci. (Preprint 1990)
[11] Kapovich, M., Millson, J.: (in preparation)
[12] Kuoroniotis, C.: Deformations of hyperbolic manifolds. Math. Proc. Camb. Philos. Soc.98, 247-261 (1985) · Zbl 0577.53041
[13] Lafontane, J.: Modules de structures conformes et cohomologie de groupes discretus, C.R. Acad. Sci., Paris, Ser. I297 (no. 13), 655-658 (1983) · Zbl 0538.53022
[14] Luo, F.: Triangulations in Moebius geometry. (To appear)
[15] Maskit, B.: Kleinian groups. Berlin Heidelberg New York: Springer 1987 · Zbl 0627.30039
[16] Newmann, W., Zagier, D.: Volumes of hyperbolicmanifolds. Topology24, 307-332 (1985) · Zbl 0589.57015
[17] Raghunathan, M.: Discrete subgroups of Lie groups. Berlin Heidelberg New York: Springer 1972 · Zbl 0254.22005
[18] Rolfsen, D.: Knots and links. Berkeley: Publish or Perish 1977 · Zbl 0339.55004
[19] Riley, R.: Nonabelian representations of 2-bridge knot groups. Q. J. Math., Oxf. II. Ser.35, 191-208 (1984) · Zbl 0549.57005
[20] Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc.15, 401-478 (1983) · Zbl 0561.57001
[21] Tan, S.: Deformations of flat conformal structures on a hyperbolic 3-manifold. J. Differ. Geom.37, 161-176 (1993) · Zbl 0785.53035
[22] Thurston, W.: Geometry and topology of 3-manifolds, Chap. 13. Princeton University Lecture Notes 1981
[23] Weil, A.: Discrete subgroups of Lie groups I, II. Ann. Math.72, 369-384 (1960);75, 578-602 (1962) · Zbl 0131.26602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.