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 $$ {\text{Mod}}_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S}}} $$ .