Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$mathrm{Ind}={ 4 cos(pi/n)^2 | n ge 3 } cup (4, infty),$$ but if we restrict to the finite depth case, the set (say $mathrm{Ind_{fd}}$) must become countable, because then the index is the squared norm of the principal graph (which then is a finite bipartite graph). According to this paper on page 63 (Afzaly-Morrison-Penneys, to appear in MAMS) there are exactly $8$ possible such indices in the interval $(4,5.25)$.

The set $mathrm{Ind_{fd}}$ is multiplicative (i.e. $alpha, beta in mathrm{Ind_{fd}} Rightarrow alpha beta in mathrm{Ind_{fd}}$) because the tensor product keeps the finite depth. Now, by this paper (Wassermann, 1998), for all $m<n$, there is a (finite depth) Jones-Wassermann subfactor of index $frac{sin^2(npi/m)}{sin^2(pi/m)}$, so that the set $mathrm{Ind_{fd}}$ has an accumultation point at $alpha n^2$ for all $n in mathbb{N}_{ge 2}$ and $alpha in mathrm{Ind_{fd}}$.

By this paper (Etingof-Nikshych-Ostrik, 2005), $mathrm{Ind_{fd}}$ is contained in the set of positive cyclotomic integers (i.e. a positive elements of $mathbb{Z}(c_n)$ with $c_n = 2cos(pi/n)$), which is a breakthrough in the understanding of $mathrm{Ind_{fd}}$.

Now, I feel like that we can get even better, because by Theorem 3.2 in this paper (Bisch, 1994): $$alpha in mathrm{Ind_{fd}} setminus mathbb{N} Rightarrow alpha^{-1} mathbb{N} cap mathrm{Ind_{fd}} = emptyset.$$

Example: $n(3-sqrt{5}) not in mathrm{Ind_{fd}}$ because $ 2frac{3+sqrt{5}}{2} cdot n(3-sqrt{5}) = 4n$, and $frac{3+sqrt{5}}{2} = 4cos^2(pi/5)$.

More strongly (see this comment) by extending Bisch’s result to every ring $R$ of cyclotomic integers (for example $mathbb{Z}(c_n)$)

$$alpha in mathrm{Ind_{fd}} setminus R Rightarrow alpha^{-1} R cap mathrm{Ind_{fd}} = emptyset.$$

The purpose of this post is not really to ask what exactly is $mathrm{Ind_{fd}}$ (which seems unreachable), but new results about it, in particular inspired by above observations.