3-Manifold Perspective on Surface Homeomorphisms for Surfaces with Very Negative Euler Characteristic

The goal of this paper is to show for a compact triangulated 3-manifold M with boundary which fibers over the circle that whenever F is a fiber with sufficiently negative Euler characteristic the monodromymaps an essential simple closed curve or an essential simple arc in F to be disjoint from its image (possibly after isotopy). This is shown by applying the theorem of Ichihara, Kobayashi, and Rieck in [10] to the double of M to get a pair of pants. We then find an equivariant pair of pants and use it to find an essential simple closed curve or an essential simple arc which satisfies our theorem. As a corollary, if we add the hypothesis that M is a hyperbolic manifold, we get that the translation distance of the monodromy in the arc and curve complex of F is at most 1 for all but finitely many monodromy maps