Almost Locally Free Groups and the Genus Question

Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F2)∩∀∃ in a firstorder language Lo appropriate for group theory. It is shown that in every model of Th(F2)∩∀∃ the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ∃-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F2)∩∀∃ while the quasi-locally free groups are the ∃-free groups with maximal Abelian subgroups elemenatarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators. 1. Preliminaries Let Lo be the first-order language whose only relation symbol is = always interpreted as the identity relation and whose only function and constant symbols are a binary operation symbol · , a unary operation symbol and a constant symbol 1. Lo shall be the language of group theory. Every sentence of Lo is logically equivalent to one in prenex normal form Q1x1 · · ·Qnxnψ(x1, ..., xn) where for each i = 1,...,n, Qi is a quantifier (∀ or ∃) and ψ(x1, ..., xn) is a formula of Lo containing no quantifiers and containing free at most the distinct variables x1, ..., xn. Vacuous quantifications are permitted and it is agreed that each of ∀xψ and ∃xψ is logically equivalent to ψ if the variable x does not occur in ψ. A sentence of Lo is both Πo and Σo if it is logically equivalent to a sentence of Lo containing no quantifiers.