On the (non) superstable part of the free group

TL;DR

This paper investigates the conditions under which definable sets in free groups are superstable, establishing a criterion that links superstability to equality of the set over finite and infinite rank free groups.

Contribution

It proves that a definable set in a free group is superstable only if it remains the same over both finite and infinite rank free groups, providing a new criterion for superstability.

Findings

Superstability implies equality of definable sets over finite and infinite free groups.

Defines a necessary condition for superstability in free groups.

Contributes to the model theory of free groups by characterizing superstable definable sets.

Abstract

In this short note we prove that a definable set $X$ over $\mathbb F_n$ is superstable only if $X(\mathbb F_n)=X(\mathbb F_{\omega})$.