Elementarily free groups are subgroup separable

@article{Wilton2005ElementarilyFG,
  title={Elementarily free groups are subgroup separable},
  author={Henry Wilton},
  journal={Proceedings of The London Mathematical Society},
  year={2005},
  volume={95},
  pages={473-496}
}
  • H. Wilton
  • Published 16 November 2005
  • Mathematics
  • Proceedings of The London Mathematical Society
Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela. 
Hall’s Theorem for Limit Groups
Abstract.A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-indexExpand
On subgroup conjugacy separability in the class of virtually free groups
A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove thatExpand
One-ended subgroups of graphs of free groups with cyclic edge groups
Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains aExpand
A Bass--Serre theoretic proof of a theorem of Burns and Romanovskii
A well known theorem of Burns and Romanovskii states that a free product of subgroup separable groups is itself subgroup separable. We provide a proof using the language of immersions and coveringsExpand
O ct 2 00 6 Hall ’ s Theorem for Limit Groups
A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. WeExpand
3-Manifold Groups
We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

References

SHOWING 1-10 OF 34 REFERENCES
SUBGROUP SEPARABILITY OF GRAPHS OF FREE GROUPS WITH CYCLIC EDGE GROUPS
It is shown that Jrl of a finite graph of free groups with cyclic edge groups is subgroup separable unless Jrl has a non-trivial element g such that gn "V gm with n i= ±m. It is easy to decide fromExpand
On Subgroup Separability in Hyperbolic Coxeter Groups
We prove that certain hyperbolic Coxeter groups are separable on their geometrically finite subgroups.
Limit groups, positive-genus towers and measure-equivalence
By definition, an $\omega$-residually free tower is positive-genus if all surfaces used in its construction are of positive-genus. We prove that every limit group is virtually a subgroup of aExpand
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Subgroups of Finite Index in Free Groups
This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total lengthExpand
Graphs and Separability Properties of Groups
Abstract A group G is LERF (locally extended residually finite) if for any finitely generated subgroup S of G and for any g  ∉  S there exists a finite index subgroup S 0 of G which contains S butExpand
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 definedExpand
Irreducible Affine Varieties over a Free Group: II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups
Abstract We shall prove the conjecture of Myasnikov and Remeslennikov [ 4 ] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements hasExpand
Irreducible Affine Varieties over a Free Group: I. Irreducibility of Quadratic Equations and Nullstellensatz
The object of this paper (which consists of two parts) is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups. We prove that anyExpand
Topology of finite graphs
This paper derives from a course in group theory which I gave at Berkeley in 1982. I wanted to prove the standard theorems on free groups, and discovered that, after a few preliminaries, the notionExpand
...
1
2
3
4
...