Realization of hyperelliptic families with the hyperelliptic semistable monodromies (Q2504126)

Let \(\Sigma _g\) be a compact real surface of genus \(g\geq 2\) without boundary. An involution \(I\) of \(\Sigma _g\) is hyperelliptic if it has \(2g+2\) fixed points. Denote by \({\mathcal M}_g\) the mapping class group of genus \(g\). Call \(\Phi \in {\mathcal M}_g\) a hyperelliptic element with \(I\) if there exists a homeomorphism \(\widetilde{\Phi}\) whose isotopy class is \(\Phi\) and which satisfies \(I\circ \tilde{\Phi}=\widetilde{\Phi}\circ I\) as a map. Denote by \([\Phi ]\) the conjugacy class of \(\Phi\) in \({\mathcal M}_g\). An element \(\Phi\) is called semistable if there exists a disjoint union of simple closed curves \({\mathcal C}:=\{ C_i\} _{i=1,2,\ldots ,r}\) and positive integers \(\{ n_i\} _{i=1,2,\ldots ,r}\) satisfying \(\Phi =D_{C_1}^{n_1}\cdots D_{C_r}^{n_r}\). Theorem. Let \(\Phi\) be a hyperelliptic semistable element. Then there exists a hyperelliptic family with monodromy \([\Phi ]\).