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Some rigidity results for coinduced actions and structural results for group von Neumann algebras

  • Author(s): Drimbe, Daniel
  • Advisor(s): Ioana, Adrian
  • et al.
Abstract

In Chapter \ref{Ch: OE} of this dissertation we prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $\Sigma$ is an infinite index subgroup of a countable group $\Gamma$, we consider a probability measure preserving (pmp) action $\Sigma\curvearrowright X_0$ and let $\Gamma\curvearrowright X$ be the coinduced action. Assume either that $\Gamma$ has property (T) or that $\Sigma$ is amenable and $\Gamma$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $\Gamma\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is cohomologous to a homomorphism from $\Gamma$ to $\mathcal V.$

We then study in Chapter \ref{Ch: prime} structural results of group von Neumann algebras arising from certain lattices following the joint work \cite{DHI16} with Daniel Hoff and Adrian Ioana. We describe all tensor product decompositions of $L(\Gamma)$ for icc countable groups $\Gamma$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(\Gamma)$ is prime, unless $\Gamma$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(\Gamma)$. As a corollary we obtain that if $\Gamma$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(\Gamma)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $\text{PSL}_2(\mathbb Z[\sqrt{d}])$ and $\text{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d\not\equiv 1\Mod{4}$ and any finite non-empty set of primes $S$.

This provides the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups.

Finally, we prove in Chapter \ref{Ch: W*} W$^*$-superrigidity for a large class of coinduced actions. We prove that if $\Sigma$ is an amenable almost-malnormal subgroup of an icc property (T) countable group $\Gamma$, the coinduced action $\Gamma\curvearrowright X$ from an arbitrary pmp action $\Sigma\curvearrowright X_0$ is W$^*$-superrigid. More precisely, if $\Lambda\car Y$ is another free ergodic pmp action such that the crossed-product von Neumann algebras are isomorphic $L^\infty(X)\rtimes\Gamma \simeq L^\infty(Y)\rtimes\Lambda$, then the actions are conjugate.

We also prove a similar statement if $\Gamma$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $\Gamma$ is W$^*$-superrigid.

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