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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).

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    $\begingroup$ I highly doubt this is known. I don't have much intuition about what being a maximal subfactor gives you, but I think the only results about fundamental groups of subfactors is what you give above. (Well almost, by Popa's work any amenable subfactor, not just finite depth, has full fundamental group, and as mention in the BNP article you cite it is true for any stable subfactor). As far as results where it is not all of $\mathbb{R}^+$, I think that the BNP examples are all that is known. $\endgroup$ Jul 11, 2013 at 22:01
  • $\begingroup$ Intuitively, a maximal subfactor is a kind of quantum generalization of a prime number, because $R^{G} \subset R$ is maximal iff $G = \mathbb{Z}/p\mathbb{Z}$ with $p$ a prime number. $\endgroup$ Jul 12, 2013 at 8:21
  • $\begingroup$ I had read about stable subfactor in the BNP: << i.e. splits a common copy of the hyperfinite $II_{1}$ factor $R$ >>, but I don't understand what does it mean. $\endgroup$ Jul 12, 2013 at 8:26
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    $\begingroup$ Stable means the that $P\subset Q\simeq P\otimes\mathcal{R}\subset Q\otimes \mathcal{R}$. $\endgroup$ Jul 13, 2013 at 4:48
  • $\begingroup$ You're just asking about subfactors of the hyperfinite II_1 here? Or do you want to let M be any factor with full fundamental group? $\endgroup$ Jul 29, 2013 at 3:00

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