The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \subset M) \} $ (see BNP)
Examples:
- It's $\mathbb{R}_{+}^{*}$ for every finite index finite depth irreducible hyperfinite subfactor (see [BNP, p262])
- It's trivial for uncountably many subfactors of the form $R^{\mathbb{Z}_{2}} \subset R⋊\mathbb{Z}_{3}$ ([BNP])
A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Examples:
- Every $2$-supertransitive subfactors: $A_{n}$-subfactor, Haagerup subfactor...
- Every group-subgroup subfactor $(R^{G} \subset R^{H})$ such that $(H \subset G)$ is a maximal subgroup
(i.e. $\pi_{H}(G)$ is a primitive permutation group with $\pi_{H} : G \to S_{X}$ canonical for $X = G/H$).
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$ ?
Remark : For being relevant, we need to restrict to factors with full fundamental group. And for being reasonable, we can start by inclusion of hyperfinite II$_{1}$ factors. Next, we could enlarge to every factors with full fundamental group (for example $L(\mathbb{F}_{\infty})$).
Reference
[BNP] Dietmar Bisch, Remus Nicoara, Sorin Popa, Continuous families of hyperfinite subfactors with the same standard invariant, International Journal of MathematicsVol. 18, No. 03, pp. 255-267 (2007).