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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant measures for algebraic actions,
Zariski dense subgroups and
Kazhdan's property $(T)$


Author: Yehuda Shalom
Journal: Trans. Amer. Math. Soc. 351 (1999), 3387-3412
MSC (1991): Primary 14L30, 20G05, 22E50, 28D15
Published electronically: April 12, 1999
MathSciNet review: 1615966
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Abstract: Let $k$ be any locally compact non-discrete field. We show that finite invariant measures for $k$-algebraic actions are obtained only via actions of compact groups. This extends both Borel's density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for $k$-algebraic actions, finitely additive finite invariant measures are obtained only via actions of amenable groups. This gives a new criterion for Zariski density of subgroups and is shown to have representation theoretic applications. The main one is to Kazhdan's property $(T)$ for algebraic groups, which we investigate and strengthen.


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Additional Information

Yehuda Shalom
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Email: yehuda@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02363-6
PII: S 0002-9947(99)02363-6
Received by editor(s): March 26, 1997
Published electronically: April 12, 1999
Additional Notes: Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).
Article copyright: © Copyright 1999 American Mathematical Society