# Definability of groups in $aleph 0$-stable metric structures - Mathematics > Logic

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Abstract: We prove that in a continuous $\aleph 0$-stable theory every type-definablegroup is definable. The two main ingredients in the proof are:\begin{enumerate} \item Results concerning Morley ranks i.e., Cantor-Bendixsonranks from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us toprove the theorem in case the metric is invariant under the group action; and\item Results concerning the existence of translation-invariant definablemetrics on type-definable groups and the extension of partial definable metricsto total ones. \end{enumerate}

Author: Itaï Ben Yaacov ICJ

Source: https://arxiv.org/