Low-slope Lefschetz fibrations

For any g = 3, we construct genus-g Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to 2. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes of all Lefschetz fibrations over the two-sphere are not realized as slopes.