Abstract

This paper produces examples of groups which provide negative answers to earlier questions of Gaglione and Spellman. One such example is a consequence of some new results on constructing n-free groups which should also be of independent interest. Specifically, the primary focus is on answering: (1) Does there exist an integer r ≥ 3 such that every finitely-generated, nonabelian r-free group is a model of the elementary theory of the non-abelian free groups? In particular, does r = 3 satisfy this condition? and (2) Let G be a non-abelian group. Is it the case that G satisfies precisely the same universal sentences as the non- abelian free groups if and only if there is an ordered abelian group Λ and a Λ-tree T such that G acts via Λ-isometries on T freely and without inversions?

Introduction and preliminaries

For a positive integer n a group G is n-free if every subgroup generated by n or fewer distinct elements is free (necessarily if rank ≤ n).

G is locally free if it is n-free for all positive integers n; moreover, following Graham Higman [15], G is countably free if every countable subgroup is free. From straightforward topological considerations an orientable surface group of genus g is (2g – 1)-free. This was generalized by B. Baumslag [2] to show that certain cyclically pinched one relator groups are 2-free and extended by Rosenberger [19] to show that these groups are also 3-free (redone in a different manner by G. Baumslag and P. Shalen [4]).