# Co‐Hopfian virtually free groups and elementary equivalence

@article{Andre2021CoHopfianVF, title={Co‐Hopfian virtually free groups and elementary equivalence}, author={Simon Andr'e}, journal={Bulletin of the London Mathematical Society}, year={2021}, volume={54} }

We prove that two co‐Hopfian finitely generated virtually free groups are elementarily equivalent if and only if they are isomorphic. We also prove that co‐Hopfian finitely generated virtually free groups are homogeneous in the sense of model theory.

## References

SHOWING 1-10 OF 22 REFERENCES

### On Tarski's problem for virtually free groups

- Mathematics
- 2019

We give a complete classification of finitely generated virtually free groups up to $\forall\exists$-elementary equivalence. As a corollary, we give an algorithm that takes as input two finite…

### Virtually free groups with finitely many outer automorphisms

- Mathematics
- 1997

Let G be a finitely generated virtually free group. From a presentation of G as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for…

### Homogeneity in the free group

- Mathematics
- 2010

We show that any non abelian free group $\F$ is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under…

### Finite Subgroups Of Hyperbolic Groups

- MathematicsInt. J. Algebra Comput.
- 2000

We prove that every finite subgroup of a hyperbolic group G can be conjugated to a 2δ+1 neighborhood of the identity element, where δ is the hyperbolicity constant for G with respect to a given…

### Trees of cylinders and canonical splittings

- Mathematics
- 2011

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively…

### Homogeneity in virtually free groups

- MathematicsIsrael Journal of Mathematics
- 2022

Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show…

### Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

- Mathematics
- 2009

This paper generalizes our work 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 over a free…

### 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…

### The Isomorphism Problem for All Hyperbolic Groups

- Mathematics
- 2010

We give a solution to Dehn’s isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary,…