Real Algebraic and Analytic Geometry |
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e-mail: , , ,
Submission: 2004, May 3.
Abstract:
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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