On the algebraic components of the SL\((2,\mathbb C)\) character varieties of knot exteriors
type: Item
instance of: scholarly article
title: On the algebraic components of the SL\((2,\mathbb C)\) character varieties of knot exteriors
zbMATH Open document ID: 0998.57017
DOI: 10.1016/S0040-9383(00)00043-4
author: Xingru Zhang
author: Steven Boyer
author: Erhard Luft
published in: Topology
publication date: 2002-06-27T00:00:00Z
review text: 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…
reviewed by: Alberto Cavicchioli
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 1761008
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: group representations
zbMATH Keywords: character variety
zbMATH Keywords: finite cyclic coverings
zbMATH Keywords: punctured torus bundles
MaRDI profile type: Publication
full work available at URL: s0040-9383(00)00043-4
OpenAlex ID: W2154095293
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description: scientific article; zbMATH DE number 1761008