An example of a property \Gamma factor with countable fundamental group

Sorin Popa

We construct a class of examples of type II$_1$ factors with the property $\Gamma$ of Murray and von Neumann but countable fundamental group. These factors do not contain any infinite dimensional property T factors and cannot be embedded into free group factors either. For the construction, we use the rigidity of the embedding $\mathbb Z^2 \subset Z^2 \rtimes\mathrm{SL}(2, \mathbb Z)$, some perturbation results for algebras that are uniformly close, and a recent result of Gaboriau on cost of measured equivalence relations implemented by $SL(2, \mathbb Z)$.