On the algebraic components of the SL\((2,\mathbb C)\) character varieties of knot exteriors
Let \(M\) be a knot exterior, that is, a compact connected orientable irreducible boundary-irreducible 3-manifold with boundary a torus. The manifold \(M\) is called hyperbolic if its interior admits a complete Riemannian metric of finite volume and constant negative sectional curvature. \(M\) is said to be small if it does not contain any closed embedded orientable surfaces which are essential, that is, incompressible and nonboundary-parallel. Given a finitely generated group \(\Gamma\), let \(R(\Gamma)\) denote the set of representations of \(\Gamma\) into \(SL(2,\mathbb{C})\) [\textit{M. Culler} and \textit{P. B. Shalen}, Ann. Math. (2) 117, 109-146 (1983; Zbl 0529.57005)]. The character of an element \(\varrho\in R(\Gamma)\) is the function \(\chi_\varrho:\Gamma\to\mathbb{C}\) definded by setting \(\chi_\varrho(\gamma) = \text{trace}(\varrho(\gamma))\). The set \(X(\Gamma)\) of the characters of the representations in \(R(\Gamma)\) is a complex affine variety, called the \(\text{SL}(2,\mathbb{C})\)-character variety of \(\Gamma\) (see the quoted paper). Set \(R(\Gamma) = R(M)\) and \(X(\Gamma) = X(M)\) if \(\Gamma\) is the fundamental group of \(M\). An algebraic component of \(X(M)\) is called nontrivial if it contains the character of an irreducible representation. If \(M\) satisfies certain conditions, the authors prove that \(M\) has finite cyclic coverings \(M_{a_i}\) with arbitrarily large numbers of nontrivial algebraic components in \(X(M_{a_i})\). Examples of knot exteriors satisfying such conditions are given by hyperbolic punctured torus bundles over the circle. Then the authors study the finite cyclic covers of the figure-eight knot exterior, and prove that for every integer \(m\) there exists a finite covering such that its \(\text{SL}(2,\mathbb{C})\)-character variety contains curve components whose associated boundary slopes have distance larger than \(m\). Finally, they show that for every integer \(m\) there exists a hyperbolic knot exterior \(M\) such that \(X(M)\) contains more than \(m\) norm curve components, each of which contains the character of a discrete faithful representation of the fundamental group of \(M\).
scientific article; zbMATH DE number 1761008
scientific article