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Uniqueness of the Group Measure Space Decomposition for Popa’s \({\mathcal{HT}}\) Factors

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Abstract

We prove that if \({\Gamma\curvearrowright (X, \mu)}\) is a free ergodic rigid (in the sense of Popa in Ann Math 163:809–889, 2006) probability measure preserving action of a group Γ with positive first \({\ell^2}\)-Betti number, then the II1 factor \({L^{\infty}(X)\rtimes\Gamma}\) has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many \({\mathcal{HT}}\) factors, including the II1 factors associated with the usual actions \({\Gamma\curvearrowright \mathbb{T^2}}\) and \({\Gamma\curvearrowright}\) \({{\rm SL}_2(\mathbb R)/{\rm SL}_2(\mathbb Z)}\), where Γ is a non-amenable subgroup of \({{\rm SL}_2(\mathbb Z)}\), have a unique group measure space decomposition.

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Adrian Ioana

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  1. Adrian Ioana
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Correspondence to Adrian Ioana.

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A. Ioana was supported by a Clay Research Fellowship.

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Ioana, A. Uniqueness of the Group Measure Space Decomposition for Popa’s \({\mathcal{HT}}\) Factors. Geom. Funct. Anal. 22, 699–732 (2012). https://doi.org/10.1007/s00039-012-0178-3

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