Stabilizers of ergodic actions of lattices and commensurators
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- by Darren Creutz and Jesse Peterson PDF
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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
- MR Author ID: 734276
- Email: creutz@usna.edu
- Jesse Peterson
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 750769
- Email: jesse.d.peterson@vanderbilt.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4119-4166
- MSC (2010): Primary 37A15; Secondary 22F10
- DOI: https://doi.org/10.1090/tran/6836
- MathSciNet review: 3624404