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On the deformation theory of representations of fundamental groups of compact hyperbolic 3-manifolds. (English) Zbl 0855.32013

The singularities of representation varieties of uniform lattices in the group \(SO (3,1)\) are studied.
A singularity is called nonquadratic, if its associated complete local ring is not isomorphic to a power series ring divided by an ideal generated by homogeneous quadratic polynomials of degree 2.
It is proved that there exists a compact hyperbolic 3-manifold \(M\) and an irreducible infinite representation \(\rho : \pi (M) \to SO(3)\) such that the singularities of the varieties \(\operatorname{Hom} (\pi (M),\;SO (3))\) at \(\rho\) and \((V (\pi_1 (M),\;SO (3)))\) at \([\rho]\) are not quadratic.
Notice that for the fundamental group of any compact Kähler manifold \(M\) and a compact Lie group \(G\) the only singularities of the representation variety \(\operatorname{Hom} (\pi_1 (M),G)\) are quadratic [cf. W. Goldman and the second author, Publ. Math., Inst. Hautes Etud. Sci. 67, 43-96 (1988; Zbl 0678.53059)].
Moreover, there exists a compact hyperbolic 3-manifold \(M\) such that for any semi-simple Lie group \(G\) the varieties \(\operatorname{Hom} (\pi_1 (M), G)\) and \(V (\pi_1 (M), G)\) have nonquadratic singularities at the trivial representations.

MSC:

32J17 Compact complex \(3\)-folds
32S30 Deformations of complex singularities; vanishing cycles
57S25 Groups acting on specific manifolds
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
58H15 Deformations of general structures on manifolds

Citations:

Zbl 0678.53059
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