Sela, Z. Diophantine geometry over groups. V\(_2\): Quantifier elimination. II. (English) Zbl 1118.20034 Geom. Funct. Anal. 16, No. 3, 537-706 (2006). The paper is the sixth in a sequence [for part IV cf. Isr. J. Math. 143, 1-130 (2004; Zbl 1088.20017)] 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. The author keeps the notions and notation introduced in the previous paper on quantifier elimination. The sieve procedure is presented that concludes the author’s analysis of definable sets, and proves quantifier elimination over a free group. It is shown that the Boolean algebra of \(AE\) sets is invariant under projection. Reviewer: V. A. Roman’kov (Omsk) Cited in 4 ReviewsCited in 10 Documents MSC: 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 Keywords:equations over groups; Makanin-Razborov diagrams; free groups; limit groups; first order theories; quantifier elimination; elementary sets; Tarski problems Citations:Zbl 1088.20017 PDFBibTeX XMLCite \textit{Z. Sela}, Geom. Funct. Anal. 16, No. 3, 537--706 (2006; Zbl 1118.20034) Full Text: DOI