Translation Distance and Fibered 3-Manifolds

A 3-manifold is said to be fibered if it is homeomorphic to a surface bundle over the circle. For a cusped, hyperbolic, fibered 3-manifold M, we study an invariant of the mapping class of a surface homeomorphism called the translation distance in the arc complex and its relation with essential surfaces in M. We prove that the translation distance of the monodromy of M can be bounded above by the Euler characteristic of an essential surface. For one-cusped, hyperbolic, fibered 3-manifolds, the monodromy can also be bounded above by a linear function of the genus of an essential surface. We give two applications of our theorems. We show that if the translation distance of the monodromy of a one-cusped, hyperbolic, fibered 3-manifold is greater than three, then every Dehn filling of the manifold is irreducible. Next we investigate Schleimer\u27s Conjecture, which states that there exists a uniform bound on translation distance of the monodromy of fibered knots. We prove that infinitely many fibered Montesinos knots satisfy Schleimer\u27s Conjecture. Lastly we prove a version of Schleimer\u27s Conjecture for certain closed, fibered braids; we show that homogeneous braids satisfying a mild hypothesis have uniformly bounded translation distance in the curve complex