On character varieties, sets of discrete characters, and nonzero degree maps
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}), and it is called \textit{elementary} if it is reducible, or is conjugate to one whose image is in \(\mathrm{PSU}(2)\) or the subgroup of \(G\) whose elements leave invariant a two point subset of \(\mathbb{C}\mathrm{P}^1\).NEWLINENEWLINEAn \textit{irreducible} manifold is one in which any embedded \((n-1)\)-sphere bounds an embedded \(n\)-ball, and an \textit{incompressible} surface in a manifold is one that is embedded so that any non-contractible loop in the surface does not bound a disk in the manifold. Then a \textit{knot manifold} is a compact, connected, orientable, irreducible 3-manifold whose boundary is an incompressible torus. Since the boundary of a knot manifold \(M\) is a torus, one defines a \textit{slope} to be a \(\partial M\)-isotopy class of essential simple closed curves; they correspond to primitive elements in \(H_1(\partial M)\). Given a slope \(\alpha\), one can define a \textit{Dehn filling} of \(M\), denoted \(M(\alpha)\), by gluing the boundary of a solid torus to \(\partial M\) using the data \(\alpha\) for the gluing map. We will call a knot manifold \textit{small} if it contains no closed essential surfaces (essential means it \(\pi_1\)-injects into the manifold and its image cannot be freely homotoped to the boundary).NEWLINENEWLINENext, an orientable, compact, connected \(3\)-manifold \(M\) \textit{dominates} another such manifold \(N\) if there is a continuous proper map from \(M\to N\) of nonzero degree (induced map \(H_3(M,\partial M)\cong \mathbb{Z}\to H_3(N,\partial N)\cong \mathbb{Z}\) is not 0). This allows for various notions of minimality to be defined. A knot manifold is \textit{minimal} if the only knot manifold it dominates is itself. A manifold is \textit{\(\mathcal{H}\)-minimal} if the only hyperbolic manifold it dominates is itself. More generally, a closed, connected, orientable 3-manifold is \textit{minimal} if the only manifold it dominates is one with finite fundamental group.NEWLINENEWLINELastly, we remind the reader of a few classes of knots. A \textit{bridge} in a knot diagram is an arc between under-crossings that has at least one over-crossing. So, a \textit{two-bridge knot} is a knot that has a diagram with exactly two bridges. Two-bridge knots are determined by reduced fractions \(p/q\) where \(p>0\) and \(q>1\). An \(n\)-\textit{twist knot} is a knot created by making \(n\) ``twists'' (\(2n\) crossings) in a closed loop and then linking the ends together. A \((p_1,...,p_n)\)-\textit{pretzel knot} is a knot that has \(n\) ``tangles'' each having \(p_i\) crossings.NEWLINENEWLINEWith terminology out of the way, the first main theorem states the following. Let \(M_{p/q}\) be the exterior of a \(p/q\) two-bridge knot, and consider the sequence: NEWLINE\[NEWLINE\pi_1(M_{p/q})\overset{\varphi_1}{\longrightarrow}\pi_1(N_1)\overset{\varphi_2}{\longrightarrow}\cdots\overset{\varphi_n}{\longrightarrow}\pi_1(N_n),NEWLINE\]NEWLINE consisting of \textit{virtual epimorphisms} (image is of finite index in its co-domain) between knot manifolds that are also non-injective. Then: (a) if \(N_i\) is small for each \(i\), then \(n<\frac{p-1}{2}\); (b) if each \(\varphi_i\) is induced by a nonzero degree map, then \(n+1\) is bounded above by the number of distinct divisors of \(p\); and (c) if each \(\varphi_i\) is induced by a degree 1 map then \(n+1\) is bounded above by the number of distinct prime divisors of \(p\).NEWLINENEWLINEAs a corollary, the authors deduce that if \(p\) is an odd prime and if \(M_{p/q}\) is hyperbolic, then \(M_{p/q}\) is minimal, and also if \(p\) is a prime power, any degree one map \(M_{p/q}\to N\), \(N\) a knot manifold, is homotopic to a homeomorphism.NEWLINENEWLINEThe second main result, concerning convergence properties in character varieties under certain conditions, implies the following corollaries:NEWLINENEWLINE(1) for \(M\) a small hyperbolic knot manifold, all but finitely many of the discrete, non-elementary, torsion-free, nonzero volume characters in \(\mathfrak{X}_0\) are induced by the complete hyperbolic structure on the interior of \(M\) or by Dehn fillings of manifolds finitely covered by \(M\); (2) let \(M\), \(N\) be two small hyperbolic knot manifolds with \(\{\alpha_n\}\) and \(\{\beta_n\}\) sequences of slopes in \(\partial M\) and \(\partial N\) respectively for which there are dominations \(M(\alpha_n)\to N(\beta_n)\), then if \(\{\alpha_n\}\) do not sub-converge projectively to the projective class of a boundary slope of \(M\), then there is a sub-sequence \(\{j\}\) of \(\{n\}\) and a domination \(M\to N\) which induces the dominations \(M(\alpha_j)\to N(\beta_j)\) for all \(j\); (3) if \(M\) is a small hyperbolic knot manifold that is \(\mathcal{H}\)-minimal and admits two non-homeomorphic Dehn fillings each of which also is \(\mathcal{H}\)-minimal, then \(M\) admits infinitely many \(\mathcal{H}\)-minimal Dehn fillings; (4) for \(M\) the exterior of a small hyperbolic knot in the 3-sphere such that \(\mu,\lambda\in H_1(\partial M)\) represent the meridional and longitudinal slopes respectively, if \(M\) is \(\mathcal{H}\)-minimal, then for all but finitely many \(n\in\mathbb{Z}\), the Dehn filled manifolds \(M(n\mu+\lambda)\) also are \(\mathcal{H}\)-minimal; and (5) for \(M\) the exterior of a hyperbolic \(\frac{p}{q}\) two-bridge knot with \(p\) prime, all but finitely many Dehn fillings of \(M\) yield \(\mathcal{H}\)-minimal manifolds.NEWLINENEWLINEThe third and final main result, which relates properties of character varieties to minimal Dehn fillings, implies the following two additional corollaries: (1) if \(M\) is the exterior of a hyperbolic twist knot, then all but finitely many Dehn fillings of \(M\) yield a minimal manifold; and (2) let \(M\) be the exterior of a hyperbolic \(\frac{p}{q}\) two-bridge knot with prime \(p\) or of a \((-2,3,n)\) pretzel knot with \(n\) not divisible by 3, then there are infinitely many slopes \(\alpha\) on \(\partial M\) such that \(M(\alpha)\) is minimal.
scientific article; zbMATH DE number 6029053