On the Geography of Slopes of Fibrations

Abstract

The slopes of fibrations of genus $g\ge 2$ have strict lower (resp., upper) bound, namely ${\lambda }_m(g)$ (resp., ${\lambda }_M(g)$). In this paper, we show that if $g\ne 3$, then each rational number $r\in [{\lambda }_m(g),{\lambda }_M(g)]$ can occur as the slope of some fibration of genus $g$. A similar result is also true for $g=3$ and $r\in [{\lambda }_m(3),9]$.