Realization of hyperelliptic families with the hyperelliptic semistable monodromies

Abstract

Let $\Phi$ be an element of the mapping class group $\mathcal{M}_{g}$ of genus $g$ ($\geq 2$) such that $\Phi$ is the isotopy class of a pseudo periodic map of negative twists. It is expected that, for each $\Phi$ which commutes with a hyperelliptic involution, there exists a hyperelliptic family whose monodromy is the conjugacy class of $\Phi$ in the mapping class group. In this paper, we give a partial solution for the conjecture in the case where $\Phi$ is a semistable element.