HNN extensions and unique group measure space decomposition of II₁ factors

Fima, Pierre; Vaes, Stefaan

doi:10.1090/S0002-9947-2012-05415-8

HNN extensions and unique group measure space decomposition of II$_1$ factors
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by Pierre Fima and Stefaan Vaes PDF

Trans. Amer. Math. Soc. 364 (2012), 2601-2617 Request permission

Abstract:

We prove that for a fairly large family of HNN extensions $\Gamma$, the group measure space II$_1$ factor $\mathrm {L}^\infty (X) \rtimes \Gamma$ given by an arbitrary free ergodic probability measure preserving action of $\Gamma$ has a unique group measure space Cartan subalgebra up to unitary conjugacy. From this we deduce new examples of W$^*$-superrigid group actions, i.e. where the II$_1$ factor $\mathrm {L}^\infty (X) \rtimes \Gamma$ entirely remembers the group action from which it was constructed.

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