\(W^*\) -Superrigidity for arbitrary actions of central quotients of braid gro

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• 作者：Ionut Chifan ; Adrian Ioana ; Yoshikata Kida
• 刊名：Mathematische Annalen
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：361
• 期：3-4
• 页码：563-582
• 全文大小：324 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807

文摘

For any \(n\geqslant 4\) let \(\tilde{B}_n=B_n/Z(B_n)\) be the quotient of the braid group \(B_n\) through its center. We prove that any free ergodic probability measure preserving (pmp) action \(\tilde{B}_n\curvearrowright (X,\mu )\) is virtually \(\hbox {W}^*\) -superrigid in the following sense: if \(L^{\infty }(X)\rtimes \tilde{B}_n\cong L^{\infty }(Y)\rtimes \Lambda \) , for an arbitrary free ergodic pmp action \(\Lambda \curvearrowright (Y,\nu )\) , then the actions \(\tilde{B}_n\curvearrowright X,\Lambda \curvearrowright Y\) are virtually conjugate. Moreover, we prove that the same holds if \(\tilde{B}_n\) is replaced with a finite index subgroup of the direct product \(\tilde{B}_{n_1}\times \cdots \times \tilde{B}_{n_k}\) , for some \(n_1,\ldots ,n_k\geqslant 4\) . The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from Popa and Vaes (212, 141-98, 2014) in combination with the OE superrigidity theorem for actions of mapping class groups from Kida (131, 99-09, 2008). Similar techniques allow us to prove that if a group \(\Gamma \) is hyperbolic relative to a finite family of proper, finitely generated, residually finite, infinite subgroups, then the \(\hbox {II}_1\) factor \(L^{\infty }(X)\rtimes \Gamma \) has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic pmp action \(\Gamma \curvearrowright (X,\mu )\) .