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Diophantine geometry over groups. VI: The elementary theory of a free group. (English) Zbl 1118.20035
The author applies the tools and techniques presented in the previous papers in the sequence [for part V\(_2\) cf. ibid. 537-706 (2006; see the preceding review Zbl 1118.20034)] to answer some of A. Tarski’s problems on the elementary theory of a free group, and generalizations of these problems.
Theorem 1. Let \(F_k\) be a (non-Abelian) free group, and let \(Q(p)\) be a definable set over \(F_k\). Then \(Q(p)\) is in the Boolean algebra of \(AE\) sets over \(F_k\).
Theorem 2. Let \(Q(p)\) be a set defined by a coefficient-free predicate over a group. Then there exists a set \(L(p)\) defined by a coefficient-free predicate which is in the Boolean algebra of \(AE\) predicates, so that for every free group \(F_k\), \(k\geq 2\), the sets \(Q(p)\) and \(L(p)\) are equivalent.
Theorem 3. The elementary theories of non-Abelian finitely generated free groups are equivalent.
Theorem 4. Let \(F_k,F_l\) be free groups for \(2\leq k\leq l\). Then the standard embedding \(F_k\to F_l\) is an elementary embedding.
Theorem 7. A finitely generated group is elementarily equivalent to a non-Abelian free group if and only if it is a non-elementary hyperbolic \(\omega\)-residually free tower.

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E05 Free nonabelian groups
20E10 Quasivarieties and varieties of groups
20A15 Applications of logic to group theory
11D72 Diophantine equations in many variables
14A22 Noncommutative algebraic geometry
03B25 Decidability of theories and sets of sentences
03C07 Basic properties of first-order languages and structures
03C60 Model-theoretic algebra
20F67 Hyperbolic groups and nonpositively curved groups
Zbl 1118.20034
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