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.
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.