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Finiteness of compact maximal flats of bounded volume

Published online by Cambridge University Press:  02 February 2004

HEE OH
HEE OH
Affiliation:
Mathematics Department, Princeton University, Princeton, NJ 08544, USA and School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: [email protected]) 253-37 Mathematics, Caltech, Pasadena, CA 91125, USA
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Abstract

Let M be a complete Riemannian locally symmetric space of non-positive curvature and of finite volume. We show that there are only finitely many compact maximal flats in M of volume bounded by a given number. As a corollary in the case $M=\mathrm{SL}_n(\mathbb{Z})\backslash\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n$, we give a different proof of a theorem of Remak that for any $n\in \mathbb{N}$, there are only finitely many totally real number fields of degree n whose regulator is less than a given number.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 1 , February 2004 , pp. 217 - 225
Copyright
2004 Cambridge University Press

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