On representations of certain pseudo-Anosov maps of Riemann surfaces with punctures

Abstract

Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\ge 1$. Let ${\alpha }_1,{\alpha }_2\subset S$ be two simple closed geodesics such that $\{{\alpha }_1,{\alpha }_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along ${\alpha }_1$ and ${\alpha }_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in $\mathcal{F}(S,a)$ that can be constructed by two filling geodesics. Finally, for any point $b\in S$, we obtain a family $\mathcal{H}$ of pseudo-Anosov maps on $S\backslash \{b\}$ that is not obtained from Thurston's construction and projects to an element $\chi \in \mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\mathcal{H}$ are also discussed.

15 pages