Stratification of \(\mathrm{SU}(r)\)-character varieties of twisted Hopf links (Q6607623)

Let \(\Gamma\) be a finitely generated discrete group, and \(H\) a Lie group. The set of group homomorphisms \(\mathrm{Hom}(\Gamma, H)\) is naturally a topological space with respect to the compact-open topology. The group \(H\) acts on \(\mathrm{Hom}(\Gamma, H)\) by conjugation; post-compose with inner automorphisms of \(H\). Let \(\mathrm{Hom}^*(\Gamma, H)\) be the subspace of \(\mathrm{Hom}(\Gamma, H)\) whose points have \textit{closed} conjugation orbits. The \textit{\(H\)-character variety of \(\Gamma\)} is then the conjugation quotient space \(\mathfrak{X}(\Gamma, H):=\mathrm{Hom}^*(\Gamma, H)/H.\)\N\NLet \(K\) be a compact Lie group. Then \(K\) is a real algebraic group and so is the zero set of a collection of real polynomials. The complex zeros of those same polynomials form a complex Lie group \(K_\mathbb{C}\). For example, if \(K=\mathrm{SU}(n)\), then \(K_\mathbb{C}=\mathrm{SL}(n,\mathbb{C})\). Lie groups \(G\) so \(G=K_\mathbb{C}\) where \(K\) is a maximal compact subgroup of \(G\) are called \textit{ reductive}.\N\NAs shown in [\textit{C. Florentino} and \textit{S. Lawton}, Contemp. Math. 590, 9--38 (2013; Zbl 1325.16010)], there exists a canonical cellular inclusion \(\iota_{\Gamma,G}:\mathfrak{X}(\Gamma, K)\hookrightarrow \mathfrak{X}(\Gamma, G)\) whenever \(G\) is reductive and \(K\) is a maximal compact subgroup in \(G\). When \(\Gamma\) is finite, this inclusion is an isomorphism, for example. In [\textit{C. Florentino} and \textit{S. Lawton}, Topology Appl. 341, Article ID 108756, 30 p. (2024; Zbl 07792388)], a finitely generated discrete group \(\Gamma\) is defined to be \(G\)-\textit{flawed} if \(\iota_{\Gamma,G}\) is a strong deformation retraction, and it is proved that \(\Gamma\) is \(G\)- flawed for \textit{ all} reductive \(G\) if \(\Gamma\) is isomorphic to a (finite) free product of (finitely generated) nilpotent groups. On the other hand, it is known that hyperbolic surface groups are \textit{never} \(G\)-flawed if \(G\) is non-abelian [\textit{I. Biswas} and \textit{C. Florentino}, Bull. Sci. Math. 135, No. 4, 395--399 (2011; Zbl 1225.14011)].\N\NThe Hopf link is two linked circles in the \(3\)-sphere \(S^3\); which has \(2\) crossings. The complement of this link deformation retracts to \(S^1\times S^1\) and so has fundamental group \(\mathbb{Z}^2\). If one twists one circle in the Hopf link around the other circle \(n\geq 1\) times, then we obtain the \textit{ twisted Hopf link} having \(2n\) crossings. The fundamental group of the complement of this link admits the following (Wirtinger) presentation: \(\Gamma_n:=\langle a, b \ |\ a^nba^{-n}b^{-1}\rangle\).\N\NThe main theorem in the paper under review establishes that all twisted Hopf link groups \(\Gamma_n\) are \(\mathrm{SL}(2,\mathbb{C})\)-flawed, and moreover \(\mathfrak{X}(\Gamma_n,\mathrm{SU}(2))\) is homotopy equivalent to \(\vee_{i=1}^n S^2\).\N\NIn the case of \(\Gamma_1\cong \mathbb{Z}^2\), we know (since abelian implies nilpotent) that \(\Gamma_1\) is \(G\)-flawed for all reductive \(G\). Given their main theorem and this base case (\(n=1\)), the authors conjecture that \(\Gamma_n\) is \(\mathrm{SL}(m,\mathbb{C})\)-flawed for all \(m\) and \(n\). The reviewer would strengthen the conjecture to this: \(\Gamma_n\) is \(G\)-flawed for all reductive \(G\).\N\NOverall the paper is well-written and interesting. Their method of proof depends on a detailed analysis of the geometry of a natural stratification of \(\mathfrak{X}(\Gamma_n, \mathrm{SU}(m))\) which may be of independent interest.\N\NFor the entire collection see [Zbl 1545.14003].