Remarks on Thurston’s Construction of Pseudo-Anosov Maps of Riemann Surfaces

Abstract

It is well known that certain isotopy classes of pseudo-Anosov maps on a Riemann surface $\tilde S$ of non-excluded type can be defined through Dehn twists $t_{\tilde \alpha }$ and $t_{\tilde \beta }$ along simple closed geodesics $\tilde \alpha$ and $\tilde \beta$ on $\tilde S$, respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane $\mathbb{H}$ so that ${\mathbb{H}}/G \cong \tilde S$. For any point a ∈ $\tilde S$ define $S = \tilde S\backslash \{ a\}$. In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on $\tilde S$ obtained from Thurston’s construction.