# On the (non) superstable part of the free group

@article{Perin2016OnT,
title={On the (non) superstable part of the free group},
author={Chlo{\'e} Perin and Rizos Sklinos},
journal={Mathematical Logic Quarterly},
year={2016},
volume={62}
}
• Published 23 November 2014
• Mathematics
• Mathematical Logic Quarterly
In this short note we prove that a definable set X over Fn is superstable only if X(Fn)=X(Fω) .
1 Citations

## References

SHOWING 1-10 OF 16 REFERENCES

### On the generic type of the free group

• R. Sklinos
• Mathematics
The Journal of Symbolic Logic
• 2011
It is proved that the set of primitive elements in finite rank free groups is not uniformly definable and that uncountable free groups are not ℵ1-homogeneous.

### Free and Hyperbolic Groups are not Equational

We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and

### Definable Sets in a hyperbolic Group

• Mathematics
Int. J. Algebra Comput.
• 2013
It is proved that proper non-cyclic subgroups of F and G are not definable and that definable subsets in a free group are either negligible or co-negligible in their terminology.

### Geometric Stability Theory

Introduction 1. Stability theory 2. The classical finite rank theory 3. Quasi finite axiomatizability 4. 1-based theories and groups 5. Groups and geometries 6. Unidimensional theories 7. Regular

### Diophantine geometry over groups VI: the elementary theory of a free group

Abstract.This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined

### On groups and fields interpretable in torsion-free hyperbolic groups

• Mathematics
• 2012
We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other

### On genericity and weight in the free group

AbstractWe prove that the generic type of the (theory of the) free group F n on n ≥ 2 generators has inﬁnite weight, strengthening the well-knownresult that these free groups are not superstable. A

### Diophantine geometry over groups VIII: Stability

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of

### Diophantine Geometry over Groups IX: Envelopes and Imaginaries

This paper is the ninth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of