On Turner's theorem and first-order theory

摘要

A theorem of E.C. Turner states that if F is a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner's characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner's theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group.