Deformations of representations of discrete subgroups of \(SO(3,1)\)
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 \(\Hom (\pi_1 (M), \text{Isom} (H^n))/ \text{Isom}_+ (H^n)\). 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.
scientific article; zbMATH DE number 683148
scientific article