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 insure 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 this paper, 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 $L(G)$ be a McDuff factor and $L^{\infty}(X, \mu) \rtimes_{\alpha} G$ being a McDuff factor?
EDIT: Does anyone have any good references on crossed products and group measure spaces?
Would MathOverflow be a better place to ask this question?