Pseudo-Anosov mapping classes and their representations by products of two Dehn twists

Abstract

Let \( \tilde S \) be a Riemann surface of analytically finite type (p, n) with 3p − 3 + n > 0. Let a ∈ \( \tilde S \) and S = \( \tilde S \) − {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on \( \tilde S \) and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on \( \tilde S \), there are infinitely many pseudo-Anosov maps F on S − {b} = \( \tilde S \) − {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.