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On Turner’s theorem and first-order theory. (English) Zbl 1392.20025

Summary: A theorem of E. C. Turner [Bull. Lond. Math. Soc. 28, No. 3, 255–263 (1996; Zbl 0852.20022)] 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.

MSC:

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
03C60 Model-theoretic algebra
20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups
03C07 Basic properties of first-order languages and structures
20E26 Residual properties and generalizations; residually finite groups
20E36 Automorphisms of infinite groups
20F67 Hyperbolic groups and nonpositively curved groups
20A15 Applications of logic to group theory

Citations:

Zbl 0852.20022
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References:

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