Slopes of fibrations with trivial vertical fundamental groups

Kodaira fibrations have non-trivial vertical fundamental groups and their slopes are all $12$. In this paper, we show that $12$ is indeed the sharp upper bound for the slopes of fibrations with trivial vertical fundamental groups. Precisely, for each g≥3 we prove the existence of fibrations of genus g with trivial vertical fundamental groups whose slopes can be arbitrarily close to $12$. This gives a relative analogy of Roulleau-Urz\'ua's work on the slopes of surfaces of general type with trivial fundamental groups.