# 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} }

In this short note we prove that a definable set X over Fn is superstable only if X(Fn)=X(Fω) .

## One Citation

## References

SHOWING 1-10 OF 16 REFERENCES

### On the generic type of the free group

- MathematicsThe 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

- Mathematics
- 2012

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

- MathematicsInt. 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

- Mathematics
- 1996

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

- Mathematics
- 2006

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

- Mathematics
- 2008

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

- Mathematics
- 2006

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

- Mathematics
- 2009

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…