Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

30 pagesGiven a family of closed Riemann surfaces with injective monodromy $E\to B$ over a manifold $B$, we explain how to build a new family of Riemann surfaces with injective monodromy whose base is a finite cover of the total space $E$ and whose fibers have higher genus. As a consequence we prove the existence of families of closed Riemann surfaces with injective monodromy whose base is an iterated Kodaira fibration of arbitrary dimension. We also prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of another closed surface. Then we address the following two questions. Can a Kodaira fibration fiber in more than two ways? When is the fundamental group of a Kodaira fibration a ${\rm CAT}(0)$ group