We examine the notion of a-strong singularity for subfactors N of a II1 factor M, which is a metric quantity that relates the distance of a unitary to a subalgebra with the distance between that subalgebra and its unitary conjugate. Using work of Popa, Sinclair, and Smith, we show that there exists an absolute constant 0 < c < 1 such that all singular subfactors are c-strongly singular. Under the hypothesis that N0 \ hM, eNi is 2-dimensional, we prove that finite index subfactors are 1-strongly singular with a constant that tends to 1 as the Jones Index tends to infinity and infinite index subfactors are 1-strongly singular. We provide examples of subfactors satisfying these conditions using group theoretic constructions. Specifically, if P is a II1 factor and G is a countable discrete group acting on P by outer automorphisms, we characterize the elements x of PoG such that x(PoH)x0 PoH for some subgroup H of G. We establish that proper finite index singular subfactors do not have the weak asymptotic homomorphism property, in contrast to the case for masas. In the infinite index setting, we discuss the role of the semigroup of one-sided normalizers with regards to the question of whether all infinite index singular subfactors have the weak asymptotic homomorphism property. Finally, we provide a characterization of singularity for finite index subfactors in terms of the traces of projections in N0 \ hM, eNi and use this result to show that fixed point subfactors of outer Zp for p prime are regular. The characterization extends to infinite index subfactors by replacing singular with contains its one-sided normalizers.

This thesis examines geometric aspects of the outer automorphism group of a finitely generate free group. Using recent advances made in understanding mapping class groups as our primary motivation, we refine methods to understand the structure of Out(F_n) via its action on free factors of F_n. Our investigation has a number of applications: First, we give a natural notion of projection between free factors, extending a construction of Bestvina-Feighn. Second, we provide a new method to produce fully irreducible automorphisms of F_n using combinations of automorphism supported on free factors. Finally, we use these results to give a general construction of quasi-isometric embeddings from right-angled Artin groups into Out(F_n). / text