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

TL;DR

This paper proves that in continuous lm-stable theories, all type-definable groups are actually definable, using Morley ranks and invariant metrics, advancing the understanding of definability in metric model theory.

Contribution

It establishes that in lm-stable metric structures, every type-definable group is fully definable, extending classical results to the continuous setting.

Findings

Type-definable groups are definable in lm-stable theories.

Morley ranks help in proving definability under invariant metrics.

Existence of translation-invariant definable metrics is crucial.

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}