Strong Singularity for Subfactors of a II 1 Factor

We examine the notion of α-strong singularity for subfactors of a II1 factor, 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 subfactors are c-strongly singular. Under additional hypotheses, we prove that certain finite index subfactors are α-strongly singular with a constant that tends to 1 as the Jones Index tends to infinity and certain infinite index subfactors are 1-strongly singular. We provide examples of subfactors satisfying these conditions using group theoretic constructions. Finally, we establish that proper finite index singular subfactors do not have the weak asymptotic homomorphism property relative to the containing factor, in contrast to the case for masas.