# McDuff groups and McDuff factors

I asked a question over on Math.Stackexchange with the same title, but I didn't get any activity over there, which made me think that the question would be better suited for MathOverflow. I suppose you members of MathOverflow will be the judge.

Let $$G$$ be a countable, infinite conjugacy class (ICC) group (i.e., if $$g \in G \setminus \{1\}$$, then the conjugacy class of $$g$$ is infinite). This will ensure that the group von Neumann algebra $$L(G)$$ is a so-called $$II_1$$ factor von Neumann algebra.

Recall that a $$II_1$$ factor von Neumann algebra $$M$$ is said to be a McDuff factor provided it tensorially absorbs the hyperfinite $$II_1$$ factor; i.e., $$M \cong M \otimes R$$, where $$R$$ is the hyperfinite $$II_1$$ factor (one way of constructing $$R$$ is to take a discrete, amenable, ICC group $$\Gamma$$ and $$L(\Gamma)$$ will be the hyperfinite $$II_1$$ factor).

In Inner amenability, property Gamma, McDuff II_1 factors and stable equivalence relations, Deprez and Vaes define a McDuff group as a group $$G$$ which admits a free ergodic probability measure preserving action $$\alpha$$ on a measure space $$(X, \mu)$$ such that the crossed product $$II_1$$ factor $$L^{\infty}(X, \mu) \rtimes_{\alpha} G$$ is a McDuff factor.

I am relatively new to this subject, so I hope my question isn't too elementary, but what is the connection/relation between these two notions? More specifically, what is the connection (if any) between $$L(G)$$ being a McDuff factor and $$L^{\infty}(X, \mu) \rtimes_{\alpha} G$$ being a McDuff factor?

• In arxiv.org/abs/1205.5123, Kida constructs an icc group $G$ that is a McDuff group (in the sense of admitting a free, ergodic, probability measure preserving action with the crossed product being a McDuff $II_1$ factor), but such that $L(G)$ does not even have property Gamma. Certainly, $L(G)$ is not a McDuff $II_1$ factor. Dec 19, 2020 at 16:21
• The converse is not clear to me and I think that it would require new ideas to construct an icc group such that $L(G)$ is a McDuff $II_1$ factor and yet, $G$ is not a McDuff group. The difficulty comes from the fact that most concrete constructions of groups $G$ with $L(G)$ being a McDuff $II_1$ factor are such that this McDuff property is witnessed by concrete central sequences living in smaller and smaller "parts" of $G$. Often you can construct an action of $G$ where these "smaller and smaller parts" asymptotically act trivially, implying that the crossed product is a McDuff $II_1$ factor. Dec 19, 2020 at 16:26