On the deformation theory of representations of fundamental groups of compact hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: On the deformation theory of representations of fundamental groups of compact hyperbolic 3-manifolds
zbMATH Open document ID: 0855.32013
DOI: 10.1016/0040-9383(95)00060-7
author: Michael Kapovich
author: John Millson
published in: Topology
publication date: 1996-09-26T00:00:00Z
review text: 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 \(\Hom (\pi (M),\;SO (3))\) 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 \(\Hom (\pi_1 (M),G)\) are quadratic [cf. \textit{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 \(\Hom (\pi_1 (M), G)\) and \(V (\pi_1 (M), G)\) have nonquadratic singularities at the trivial representations.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 58H15
zbMATH DE Number: 929564
zbMATH Keywords: compact hyperbolic 3-manifold
zbMATH Keywords: fundamental group
MaRDI profile type: Publication
full work available at URL: 0040-9383(95)00060-7
OpenAlex ID: W1999472475
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description: scientific article; zbMATH DE number 929564