A classification problem on mapping classes on fiber spaces over Teichmüller spaces

Abstract

Let $\tilde{S}$ be an analytically finite Riemann surface which is equipped with a hyperbolic metric. Let $S=\tilde{S}\backslash \{\mbox{one point}\ x\}$. There exists a natural projection $\Pi$ of the $x$-pointed mapping class group Mod$_S^x$ onto the mapping class group Mod$(\tilde{S})$. In this paper, we classify elements in the fiber $\Pi^{-1}(\chi)$ for an elliptic element $\chi\in \mbox{Mod}(\tilde{S})$, and give a geometric interpretation for each element in $\Pi^{-1}(\chi)$. We also prove that $\Pi^{-1}(t_a^n\circ \chi)$ or $\Pi^{-1}(t_a^n\circ \chi^{-1})$ consists of hyperbolic mapping classes provided that $t_a^n\circ \chi$ and $t_a^n\circ \chi^{-1}$ are hyperbolic, where $a$ is a simple closed geodesic on $\tilde{S}$ and $t_a$ is the positive Dehn twist along $a$.