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

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) 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. We apply our construction to prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of a closed surface of higher genus.
Comment: 32 pages, v3. The order of the sections has changed. This is the final version, to be published by Math. Annalen