On the local structure of \(SL(2,{\mathbb C})\)-character varieties at reducible characters
Let \(M\) be a compact connected irreducible orientable 3-manifold whose boundary is an incompressible torus, and \(\Gamma\) be the fundamental group of \(M\). The \(SL(2,\mathbb{C})\)-representation variety of \(M\) is a complex affine algebraic variety \(R(M)\) whose points correspond to the representations of \(\Gamma\) in \(SL(2,\mathbb{C})\). The quotient of \(R(M)\) by the natural action of \(SL(2,\mathbb{C})\) is the character variety which is denoted by \(X(M)\). The author studies the local behavior of the projection \(R(M)\to X(M)\) at reducible representations, and describes a 1-cocyle condition which guarantees the smoothness of a reducible character in \(X(M)\). The author applies the results mentioned above and the results of [\textit{L. Ben Abdelghani} and \textit{S. Boyer}, Proc. Lond. Math. Soc. (3) 83, No. 1, 235-256 (2001; Zbl 1029.57001)] to exceptional Dehn fillings. In particular, he proves the following statement: Let \(K\) be a hyperbolic knot in \(S^3\) with exterior \(M\); then there is at most one slope \(r\) on the boundary \(\partial M\) of \(M\) such that the corresponding manifold \(M(r)\) obtained by Dehn filling is a Seifert fibered space whose base orbifold is Euclidean; if there is such a slope, it is integral, though not longitudinal.
scientific article; zbMATH DE number 1787535
scientific article