N ov 2 00 5 Elementarily free groups are subgroup separable Henry Wilton 16 th November 2005

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. Limit groups arise naturally in the study of the set of homomorphisms to free groups and, in the guise of fully residually free groups, have long been studied in connection with the first-order logic of groups: see for, example, [11], [12] and [15]. Indeed, limit groups turn out to be precisely the groups with the same existential theory as a free group [15]. The name limit group was introduced by Zlil Sela in his solution to the Tarksi Problem (see [22], [23] et seq.), wherein he characterized the finitely generated groups with the same elementary theory as a free group. Such group are called elementarily free. Sela asked if limit groups were subgroup separable in [20]. A group G is subgroup separable (or LERF ) if any finitely generated subgroup H and any element g / ∈ H can be distinguished in a finite quotient. The generalized word problem is soluble for subgroup separable groups. Historically, it has been of interest in topology because, in certain circumstances, an immersion into a space with subgroup separable fundamental group can be lifted to an embedding in a finite cover (see, for example, [26]). The first significant result concerning subgroup separability is due to M. Hall in [8], who demonstrated the property for free groups. This was The existential theory of G is the set of sentences in the elementary theory that only use the existential quantifier ∃. The elementary theory of a group G is the set of first-order sentences that are true in G.