Integral points on character varieties
type: Item
instance of: scholarly article
title: Integral points on character varieties
zbMATH Open document ID: 1048.57009
published in: Mathematische Annalen
publication date: 2003-06-02T00:00:00Z
review text: Let \(M\) be a finite volume hyperbolic \(3\)-manifold with a single cusp, and let \(X(M)\) be the character variety associated to representations of its fundamental group to \(SL(2,\mathbb{C})\). It is known that the component \(X_0\) of \(X(M)\) containing the character of the faithful discrete representation is an irreducible algebraic curve, and that \(X_0\) carries much topological information about \(M\). For a quadratic imaginary number field \(\mathbb{Q}(\sqrt{-d})\), let \(O_d\) denote the ring of integers. A point on \(X_0\) whose coordinates are in \(\mathbb{Z}\) (resp. \(O_d\)) is called a \(\mathbb{Z}\)-integral point (resp. \(O_d\)-point). The authors prove that the number of \(O_d\)-points on \(X_0\) is finite for each \(d\), and that for large enough \(d\), there are no \(O_d\)-points on \(X_0\) other than possibly \(\mathbb{Z}\)-integral points. For \(\mathbb{Z}\)-points, they can say more. If a hyperbolic knot \(K\) in an integral homology \(3\)-sphere admits a \(\mathbb{Z}\)-integral point on its \(X_0\), then the point corresponds to a surjection of the knot group onto either \(\Sigma_3^*\) or \(A_4^*\), where \(\Sigma_3^*\) or \(A_4^*\) denote the central extensions by lifting the symmetric group on \(3\) letters and alternating group on \(4\) letters from \(SO(3)\) to \(SU(2)\). In fact, the existence of such surjection is controlled by Alexander polynomials. As a stronger result, in case that \(X(K)\) contains a \(\mathbb{Z}\)-integral point corresponding to an irreducible representation, the same conclusion holds, or the knot group surjects onto the trefoil knot group. Hence, the number of integral points on \(X(K)\) corresponding to an irreducible representation is finite, unless the knot group surjects onto the trefoil group. There are also examples of calculations and an application to finite Dehn surgery problems on hyperbolic knots in \(S^3\).
reviewed by: Masakazu Teragaito
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 1922399
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: integral point
zbMATH Keywords: character variety
zbMATH Keywords: Dehn surgery
author: D. D. Long
author: Alan W. Reid
MaRDI profile type: Publication
full work available at URL: s00208-002-0380-y
OpenAlex ID: W2146142712
DOI: 10.1007/S00208-002-0380-Y
description: scientific article; zbMATH DE number 1922399