Real Algebraic and Analytic Geometry

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103. Alessandro Berarducci, Margarita Otero, Ya'acov Peterzil, Anand Pillay:
A descending chain condition for groups definable in o-minimal structures.

e-mail: , , ,

Submission: 2004, May 3.

We prove that if $G$ is a group definable in a saturated $o$-minimal structure, then $G$ has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, $G$ has a smallest (necessarily normal) type-definable subgroup $G^{00}$ of bounded index and $G/G^{00}$ equipped with the ``logic topology" is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

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