Orbit of the diagonal in the power of a nilmanifold

Author:
A. Leibman

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1619-1658

MSC (2000):
Primary 37C99; Secondary 22E25

Published electronically:
October 20, 2009

MathSciNet review:
2563743

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Abstract: Let be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group , and let . We study the closure of the orbit of the diagonal of under the action , where are integer-valued polynomials in integer variables. (Knowing this closure is crucial for finding limits of the form , where is a measure-preserving transformation of a finite measure space and are subsets of , and limits of the form , where are subsets of **Z** and is the density of in **Z**.) We give a simple description of the closure of the orbit of the diagonal in the case that all are linear, in the case that is connected, and in the case that the identity component of is commutative; in the general case our description of the orbit is not explicit.

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Additional Information

**A. Leibman**

Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Email:
leibman@math.ohio-state.edu

DOI:
https://doi.org/10.1090/S0002-9947-09-04961-7

Keywords:
Orbits on nilmanifolds,
complexity of polynomial systems

Received by editor(s):
May 27, 2008

Published electronically:
October 20, 2009

Additional Notes:
This research was supported in part by NSF grants DMS-0345350 and DMS-0600042.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.