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arXiv · 0802.4286

Definability of groups in $\aleph_0$-stable metric structures

Itaï Ben Yaacov
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Abstract

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate}

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Itaï Ben Yaacov. 2008-02-28. Definability of groups in $\aleph_0$-stable metric structures. https://doi.org/10.2178/jsl/1278682202

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