Bounds on degrees of covers with injective monodromy in iterated Kodaira fibrations

Abstract

Let $\pi :X\to Y$ be an $n$-dimensional iterated Kodaira fibration with fiber of genus $g$ and injective monodromy. In [6], it is proved that we can pass to a finite index subgroup of ${\pi }_1(X)$ to get the base space of an ($n+1$)-dimensional iterated Kodaira fibration with injective monodromy, and they asked about bounding the index of such a group. We provide a bound on this index.