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

We study several geometric and group theoretical problems related to Kodaira fibrations, more general families of Riemann surfaces, surface-by-surface groups. First we provide constraints on fibrations that fiber in than two distinct ways, addressing a question by Catanese Salter about their existence. Then show if the fundamental surface bundle over is ${\rm CAT}(0)$ group, must have injective monodromy (unless has finite image). Finally, given family closed surfaces (of genus $\ge 2$) with $E\to B$ manifold $B$, explain how build new whose base cover total space $E$ fibers higher genus. apply our construction prove mapping class once punctured virtually admits irreducible morphisms into