Metabelian \(\mathrm{SL}(N,\mathbb{C})\) representations of knot groups. III: Deformations (Q2922451)

Let \(K\) be an oriented knot whose Alexander polynomial has a zero that is not a root of unity, \(N_K\) the complement of \(K\) in an integral homology 3-sphere, and \(\pi\) the fundamental group of \(N_K\). A representation \(\rho:\pi\to \mathrm{SL}(n,\mathbb{C})\) is called metabelian if \(\rho\) is trivial upon restriction to the second commutator subgroup of \(\pi\). The \(\mathrm{SL}(n,\mathbb{C})\)-character variety of \(\pi\) is the Geometric Invariant Theory quotient \(\mathfrak{X}_n:=\mathrm{Hom}(\pi,\mathrm{SL}(n,\mathbb{C}))/\!/\mathrm{SL}(n,\mathbb{C})\) with respect to the conjugation action of \(\mathrm{SL}(n,\mathbb{C})\) on \(\mathrm{Hom}(\pi,\mathrm{SL}(n,\mathbb{C}))\). There is a natural quotient morphism \(q:\mathrm{Hom}(\pi,\mathrm{SL}(n,\mathbb{C}))\to \mathfrak{X}_n\). A representation is irreducible if it has a closed orbit and its \(\mathrm{PSL}(n,\mathbb{C})\)-stabilizer is finite. A point in \(\chi\in\mathfrak{X}\) will be call irreducible (respectively, metabelian) if \(q^{-1}(\chi)\) contains an irreducible (respectively, metabelian) representation. Note that each irreducible point in \(\mathfrak{X}_n\) corresponds to a unique conjugation orbit of an irreducible representation \(\rho\in \mathrm{Hom}(\pi,\mathrm{SL}(n,\mathbb{C}))\); as such, we denote irreducible points by \(\chi_\rho\). The first theorem in this interesting and well-written paper shows that the number of irreducible metabelian points in \(\mathfrak{X}_n\), although finite for any given \(n\), increases exponentially as \(n\to \infty\). This implies that for all but perhaps finitely many \(n\)'s, irreducible metabelian points exist. For any irreducible \(\chi_\rho\), \(H^1(N_K;\mathfrak{sl}(n,\mathbb{C})_{\mathrm{Ad}\rho})\) linearly maps to the tangent space \(T_{\chi_\rho}(\mathfrak{X}_n)\). With this in mind, the next main result shows that the dimension of \(H^1(N_K;\mathfrak{sl}(n,\mathbb{C})_{\mathrm{Ad}\rho})\) is always greater than or equal to \(n-1\) for any irreducible metabelian character \(\chi_\rho\in \mathfrak{X}_n\). Moreover, when the dimension is equal to \(n-1\) it is shown that \(\chi_\rho\) is a simple point in \(\mathfrak{X}_n\) (contained in an unique algebraic component and smooth in that component), and therefore there exists a smooth complex \((n-1)\)-dimensional deformation of \(\chi_\rho\). Under this same hypothesis, it is then shown that \(\rho\) has finite image and therefore \(\chi_\rho\) is in fact in \(\mathfrak{Y}_n:=\mathrm{Hom}(\pi, \mathrm{SU}(n))/\mathrm{SU}(n)\hookrightarrow \mathfrak{X}_n\). Thus, \(\chi_\rho\) can be taken to be the unique metabelian representative in the aforementioned deformation. It is noted that there also exists a \textit{real} \((n-1)\)-dimensional deformation in \(\mathfrak{Y}_n\). Lastly, when the Alexander polynomial has no zero which is a root of unity, a topological criterion is described that ensures the cohomological dimension of \(n-1\) is achieved.