It is wellknown that that the fundamental group (in the sense of Murray and von Neumann) of the factor $L(F_{\mathbb{N}})$ is $\mathbb{R} \smallsetminus \{0\}$. I think that by the cutting and pasting technique and random matrix model, it is not hard to show that the fundamental group of $L(F_{S})$ is $\mathbb{R} \smallsetminus \{0\}$, where $S$ is an infinite set (not countable) and $F_S$ is the free group over $S$. Can anyone point me to some reference includes this kind of result? Thank you in advance!

4$\begingroup$ I don't know about others, but I, for one, have no clue what you are talking about. Can you please define your terms? $\endgroup$ – Igor Rivin Sep 1 '17 at 14:42

2$\begingroup$ $L(G)$ denotes the von Neumann algebra of a group $G$. This is standard terminology for von Neumann algebraists. "Factor" is in the sense of von Neumann. I agree that I was initially as stuck as Igor, given that not even the word "von Neumann algebra" was written by the OP... $\endgroup$ – YCor Mar 15 '18 at 22:04

3$\begingroup$ To a von Neumann algebra (at least those called $II_1$factors) is associated a certain subgroup of $\mathbf{R}_{>0}$ called its "fundamental group". This was defined by Murray and von Neumann. Probably they had no more imagination this very day and now this is widespread terminology. It's unrelated to topologists's fundamental group, it's not even a group (but a subgroup of $\mathbf{R}_{>0}$  sure it's a group but what's important is to remember which subgroup it is, not just the isomorphism class). $\endgroup$ – YCor Mar 15 '18 at 22:54

$\begingroup$ For uses of "fundamental group" that is not Poincaré's original topological one, I tend (especially out of context) to use an additional name. For instance "BassSerre fundamental group". I'd see nothing against using, at least out of context (e.g., in the title of such a post), "Murrayvon Neumann fundamental group". $\endgroup$ – YCor Mar 16 '18 at 6:13
This is done in the following paper:
F. Rădulescu. The fundamental group of the von Neumann algebra of a free group with infinitely many generators is $\mathbb{R}\smallsetminus\{0\}$. J. Amer. Math. Soc. 5(3) (1992), 517532.
J. AMS link (unrestricted). Other link (pdfs.semanticscholar.org)

$\begingroup$ Thanks Koushik! However, Radulescu only showed the fact for separable von Neumann algebra in that paper. $\endgroup$ – heller Mar 20 '18 at 6:14