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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.

##### MSC:
 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.)
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