  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.
