Cocycle Superrigidity for Profinite Actions of Property (T) Groups

By Adrian Ioana. 2008

91 citations recorded by the source corpus.

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that $\Gamma$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $\Gamma\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:\Gamma\times K\rightarrow\Delta$ with values in a countable group $\Delta$ is cohomologous to a cocycle which factors through the map $\Gamma\times K\rightarrow\Gamma\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $\Gamma=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually cocycle superrigid and virtually W$^*$-superrigid, for any finite nonempty set of primes $S$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94543c199b5a508c5bf96cf81b048f0ed10b6dc1

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

W*-superrigidity for Bernoulli actions of property (T) groups (2010, methodology): “Prior to these results, several large classes of OE– superrigid actions were found ([11], [37], [39], [21], [15], [22]).”
Topological stability and pseudo-orbit tracing property of group actions (2016, background): “Profinite actions have been studied extensively in rank gradient, orbit equivalence, operator algebras, and sofic entropy theory [2, 5, 7].”
Global Aspects of Ergodic Group Actions (2010, background): “[I2] A. Ioana, Cocycle superrigidity for profinite actions of property (T) groups, arXiv e-print, math.”