Transactions of the American Mathematical Society

Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property

Author(s): Yehuda Shalom
Journal: Trans. Amer. Math. Soc. 351 (1999), 3387-3412.
MSC (1991): Primary 14L30, 20G05, 22E50, 28D15
Posted: April 12, 1999
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Abstract: Let

be any locally compact non-discrete field. We show that finite invariant measures for

-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

-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

for algebraic groups, which we investigate and strengthen.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14L30, 20G05, 22E50, 28D15

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: 10.1090/S0002-9947-99-02363-6
PII: S 0002-9947(99)02363-6
Received by editor(s): March 26, 1997
Posted: April 12, 1999
Additional Notes: Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).
Copyright of article: Copyright 1999, American Mathematical Society

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