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...

We describe a simple locally CAT(0) classifying space for XXL type Artin groups (with all labels at least 5). Furthermore, when the group is not dihedral, we rank 1 periodic geodesic, thus proving that are acylindrically hyperbolic. Together with Property RD proved by Ciobanu, Holt and Rees, property implies Baum–Connes conjecture groups.