Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Abstract

Let $F^{\prime },F$ be any two closed orientable surfaces of genus $g^{\prime }>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a pseudo-Anosov map $f^{\prime }:F^{\prime }\to F^{\prime }$ so that there is a fiber preserving degree one map $M(F^{\prime },f^{\prime })\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f^{\prime }$ can be chosen so that the surface bundles $M(F^{\prime },f^{\prime })$ and $M(F,f)$ have the same first Betti numbers.

10 pages, 3 figures