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Stabilizers of ergodic actions of lattices and commensurators


Authors: Darren Creutz and Jesse Peterson
Journal: Trans. Amer. Math. Soc. 369 (2017), 4119-4166
MSC (2010): Primary 37A15; Secondary 22F10
DOI: https://doi.org/10.1090/tran/6836
Published electronically: November 8, 2016
MathSciNet review: 3624404
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Abstract: We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices.

The above are derived from more general results on groups with the Howe-Moore property and property $ (T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.


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Additional Information

Darren Creutz
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Address at time of publication: Department of Mathematics, US Naval Academy, Annapolis, Maryland 21402
Email: creutz@usna.edu

Jesse Peterson
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: jesse.d.peterson@vanderbilt.edu

DOI: https://doi.org/10.1090/tran/6836
Received by editor(s): March 31, 2014
Received by editor(s) in revised form: June 10, 2015
Published electronically: November 8, 2016
Additional Notes: This work was partially supported by NSF Grant 0901510 and a grant from the Alfred P. Sloan Foundation
Article copyright: © Copyright 2016 American Mathematical Society