Doctor of Philosophy

Abstract

dissertationA mapping class group element can be understood by an inductive process - by passing to its action on the curve complexes of the subsurfaces in the complement of the curves it fixes. By the result of Masur and Minsky, the curve complex of any surface of finite type is hyperbolic. A fully irreducible outer automorphism ($\operatorname{Out}(\mathbb{F})$ analog of a pseudo-Anosov) acts with positive translation length on the free factor complex, which is also a hyperbolic space. But a reducible outer automorphism $\Phi$ fixes the invariant free factor $A$ in the free factor complex and thus, the action is not very informative. In analogy to subsurfaces, we then look at the action of $\Phi$ on the free factor complex relative to $A$, which is a hyperbolic complex that captures the information in the complement of $A$. In this dissertation, we prove that a fully irreducible outer automorphism relative to a free factor system $\mathcal{FFA}$ acts with positive translation length on the free factor complex relative to $\mathcal{FFA}$. In order to prove this, we prove the following key results: 1. Define relative currents and prove that $\Phi$ acts with uniform north-south dynamics on a certain subspace of the space of projectivized relative currents. 2. $\Phi$ acts with uniform north-south dynamics on the closure of relative outer space. 3. Define an intersection form between the space of projective relative currents and the closure of relative outer space