Let $N\subset M$ be an inclusion of ${\rm II}_1$ factors of finite index, $[M:N]<\infty$. I would be mostly interested in the hyperfinite case, $N\simeq M\simeq R$, but let us just take them arbitrary.

There is an evolving theory about "what can be said about $N\subset M$, in the general case'', which started with Jones' index theorem, in 1983. Well-known results here include the Pimsner-Popa basis and entropy formula, the bimodule interpretation, the planar algebra formalism.

- Question: assuming that we are still in the general case, but with the extra assumption that the index is an integer, $[M,N]\in\mathbb N$, what else can be said about $N\subset M$?

To my knowledge, at least some time ago (5-10 years), the only answer here was just that the Pimsner-Popa basis is a "clean" one, I mean as in standard linear algebra. I was wondering if any advances on this question come from the recent work on subject, in small or arbitrary index I mean, perhaps as some corollaries of the theory developed there (?) I would be interested in any comment/answer here, this is actually a question that I spent some time on, long time ago.