On character varieties, sets of discrete characters, and nonzero degree maps
type: Item
instance of: scholarly article
title: On character varieties, sets of discrete characters, and nonzero degree maps
zbMATH Open document ID: 1257.57021
author: Michel Boileau
author: Steven Boyer
published in: American Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0701384
full work available at URL: Boyer-2012a-preprint.pdf
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 6029053
zbMATH Keywords: fundamental group
zbMATH Keywords: 3-manifold
zbMATH Keywords: hyperbolic
zbMATH Keywords: character variety
zbMATH Keywords: domination
zbMATH Keywords: minimal knot manifold
zbMATH Keywords: Dehn filling
MaRDI profile type: Publication
OpenAlex ID: W2040306612
arXiv ID: math/0701384
DOI: 10.1353/AJM.2012.0013
review text: The paper begins with the sentence: ``Character variety methods have proven an essential tool for the investigation of problems in low-dimensional topology and have been instrumental in the resolution of many well-known problems.'' This well-written and interesting paper contributes to this thematic sentence in content, method, and consequence.NEWLINENEWLINEThis review will summarize some of the main results, or their corollaries, as outlined in the introduction. Most of these results follow from more general and at times more technical results in the main body of the paper. These main results concern character varieties, hyperbolic knot manifolds, and nonzero degree maps. We start by reminding the reader of some terminology.NEWLINENEWLINEThroughout this review \(G=\mathrm{PSL}_2(\mathbb{C})\), and \(\Gamma=\pi_1(M)\) is the fundamental group of a connected, compact, orientable \(3\)-manifold \(M\). The set of homomorphisms \(\mathrm{Hom}(\Gamma, G)\) is an algebraic set where \(G\) acts rationally by conjugation. The GIT quotient by this action \(\mathrm{Hom}(\Gamma, G)/\!/G\) is called the \(G\)-character variety of \(\Gamma\). Denote it by \(\mathfrak{X}_\Gamma(G)\). A \textit{principal component} in \(\mathfrak{X}_\Gamma(G)\), denoted by \(\mathfrak{X}_0\), is defined to be an irreducible component containing an equivalence class of a discrete, faithful, irreducible representation. When \(M\) is hyperbolic with boundary, to each representation \(\rho:\Gamma\to G\) there is a \textit{volume} defined by taking a pseudo-developing map from the universal cover of \(M\) to hyperbolic 3-space, pulling-back the hyperbolic volume form, and integrating. It is \(G\)-invariant and does not depend on the choice of developing map as long as all such maps agree on the boundary of \(M\); this occurs for classes in \(\mathfrak{X}_0\). A representation is called reducible if its image is conjugate to the set of upper triangular matrices (otherwise it is \textit{irreducible}), an…
reviewed by: Sean Lawton
description: scientific article; zbMATH DE number 6029053