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Espace des représentations du groupe d’un noeud classique dans un groupe de Lie. (Representation space of classical knot groups in a Lie group). (French) Zbl 0956.57006
Ann. Inst. Fourier 50, No. 4, 1297-1321 (2000); addendum ibid. 51, No. 4, 1151-1152 (2001).
Let \(X\) be the complement of a knot in a rational homology 3-sphere, let \(\Pi\) be the fundamental group of \(X\) and let \(\rho_0\) be an abelian representation of \(\Pi\) in a connected complex reductive Lie group \(G\) which factorizes through the infinite cyclic group \(H_1(X,Z)/\text{tors}(H_1(X,Z))\). In this paper, the author proposes a general method for the study of the local structure of the variety of representations of \(\Pi\) in a neighborhood of the representation \(\rho_0\). In particular, she proves the following
Theorem. Suppose that the group \(\Pi\) is equipped with an involution \(k\) such that \(k(\mu)=\mu^{-1}\), where \(\mu\) is a meridian of the knot, and suppose that \(\rho_0(\mu)=h_0\), where \(h_0\) is in the Cartan subalgebra \(\mathcal H\) of \(\mathcal G\). Suppose also that there exists a root \(\alpha_{i_{0}}\) of the Lie algebra \(\mathcal G\) with respect to the Cartan subalgebra \(\mathcal H\) such that \(w_{i_{0}}= e^{{\alpha_{i_{0}}}(h_{0})}\) is a simple root of the Alexander polynomial of the knot. Then, there exists an arc of non-metabelian representations \(\rho_t:\Pi\to G\) having \(\rho_0\) as an endpoint.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20C99 Representation theory of groups
57N10 Topology of general \(3\)-manifolds (MSC2010)
Full Text: DOI Numdam Numdam EuDML
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