Outer automorphism groups of some equivalence relations

• Autores: Alex Furman
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 80, Nº 1, 2005, págs. 157-196
• Idioma: inglés
• DOI: 10.4171/cmh/10
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let $\Rel$ a be countable ergodic equivalence relation of type ${\rm II}_1$ on a standard probability space $(X,\mu)$. The group $\Rout\Rel$ of \emph{outer automorphisms} of $\Rel$ consists of all invertible Borel measure preserving maps of the space which map $\Rel$-classes to $\Rel$-classes modulo those which preserve almost every $\Rel$-class. We analyze the group $\Rout\Rel$ for relations $\Rel$ generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of $\Rout\Rel$ and explicitly computing $\Rout\Rel$ for the standard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction.