Original Paper

Geometric & Functional Analysis GAFA

, Volume 16, Issue 3, pp 537-706

First online:

Diophantine geometry over groups V2: quantifier elimination II

  • Z. SelaAffiliated withInstitute of Mathematics, Hebrew University Email author 

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This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of an AE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra of AE sets defined over a free group is invariant under projections, hence, show that every elementary set defined over a free group is in the Boolean algebra of AE sets. The procedures we use for quantifier elimination, presented in this paper, enable us to answer affirmatively some of Tarski’s questions on the elementary theory of a free group in the last paper of this sequence.

Keywords and phrases.

Equations over groups Makanin–Razborov diagrams free groups limit groups first order theory quantifier elimination Tarski problems

AMS Mathematics Subject Classification.

20F65 03BB25 20E05 20F10