## 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

### Keywords:

compact hyperbolic 3-manifold; fundamental group

Zbl 0678.53059
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