Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Measurable rigidity of actions on infinite measure homogeneous spaces, II

Author: Alex Furman
Journal: J. Amer. Math. Soc. 21 (2008), 479-512
MSC (2000): Primary 37A17; Secondary 37A35, 22E40, 22F30
Published electronically: December 27, 2007
MathSciNet review: 2373357
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Abstract: We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups $ \Gamma<G$ on homogeneous spaces $ G/H$ where $ H$ is a sufficiently rich unimodular subgroup in a semi-simple group $ G$. We also consider actions of discrete groups of isometries $ \Gamma<\mathrm{Isom}(X)$ of a pinched negative curvature space $ X$, acting on the space of horospheres $ \mathrm{Hor}(X)$. For such systems we prove that the only measurable isomorphisms, joinings, quotients, etc., are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger but uses completely different techniques which lead to more general results.

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Additional Information

Alex Furman
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607

Received by editor(s): March 8, 2006
Published electronically: December 27, 2007
Additional Notes: The author was supported in part by NSF grant DMS-0094245 and BSF USA-Israel grant 2004345.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.