A CLASSIFICATION PROBLEM ON MAPPING CLASSES ON FIBER SPACES OVER TEICHMÜLLER SPACES

Let S̃ be an analytically finite Riemann surface which is equipped with a hyperbolic metric. Let S = S̃ \{one point x}. There exists a natural projection Π of the x-pointed mapping class group Modx S onto the mapping class group Mod(S̃ ). In this paper, we classify elements in the fiber Π−1(χ) for an elliptic element χ ∈ Mod(S̃ ), and give a geometric interpretation for each element in Π−1(χ). We also prove that Π−1(tn a ◦ χ) or Π−1(tn a ◦ χ−1) consists of hyperbolic mapping classes provided that tn a ◦ χ and tn a ◦ χ−1 are hyperbolic, where a is a simple closed geodesic on S̃ and ta is the positive Dehn twist along a.