## Abstract

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.

Original language | English |
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Pages (from-to) | 537-706 |

Number of pages | 170 |

Journal | Geometric and Functional Analysis |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2006 |

### Bibliographical note

Funding Information:Keywords and phrases: Equations over groups, Makanin–Razborov diagrams, groups, limit groups, first order theory, quantifier elimination, Tarski problems. AMS Mathematics Subject Classification: 20F65 (03BB25, 20E05, 20F10) Partially supported by an Israel Academy of Sciences Fellowship.

## Keywords

- Equations over groups
- First order theory
- Free groups
- Limit groups
- Makanin-Razborov diagrams
- Quantifier elimination
- Tarski problems

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