Hyperbolic Mapping Classes and Their Lifts on the Bers Fiber Space

Abstract

Let S be a Riemann surface with genus p and n punctures. Assume that 3p−3+n > 0 and n ≥ 1. Let a be a puncture of S and let $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S} = S \cup {\left\{ a \right\}}$. Then all mapping classes in the mapping class group Mod _S that fixes the puncture a can be projected to mapping classes of ${\text{Mod}}_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S}}} $ under the forgetful map. In this paper the author studies the mapping classes in Mod _S that can be projected to a given hyperbolic mapping class in ${\text{Mod}}_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S}}} $.

Keywords

Riemann surfaces / Absolutely extremal Teichmüller mapping / Mapping classes / Teichmüller spaces / Bers fiber spaces / 30F40 / 32G05

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