Abstract
In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.
The first author was partially supported by NSF grant #MCS77-24103, the second by NSF grant #MCS-8200639.
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Johnson, D., Millson, J.J. (1987). Deformation Spaces Associated to Compact Hyperbolic Manifolds. In: Howe, R. (eds) Discrete Groups in Geometry and Analysis. Progress in Mathematics, vol 67. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6664-3_3
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