De Gruyter Expositions in Mathematics -- Title Page -- Copyright Page -- Preface -- Table of Contents -- 1 Group theory and logic: introduction -- 1.1 Group theory and logic -- 1.2 The elementary theory of groups -- 1.3 Overview of this monograph -- 2 Combinatorial group theory -- 2.1 Combinatorial group theory -- 2.2 Free groups and free products -- 2.3 Group complexes and the fundamental group -- 2.4 Group amalgams -- 2.5 Subgroup theorems for amalgams -- 2.6 Nielsen transformations -- 2.7 Bass-Serre theory -- 3 Geometric group theory -- 3.1 Geometric group theory -- 3.2 The Cayley graph -- 3.3 Dehn algorithms and small cancellation theory -- 3.4 Hyperbolic groups -- 3.5 Free actions on trees: arboreal group theory -- 3.6 Automatic groups -- 3.7 Stallings foldings and subgroups of free groups -- 4 First order languages and model theory -- 4.1 First order language for group theory -- 4.2 Elementary equivalence -- 4.3 Models and model theory -- 4.4 Varieties and quasivarieties -- 4.5 Filters and ultraproducts -- 5 The Tarski problems -- 5.1 The Tarski problems -- 5.2 Initial work on the Tarski problems -- 5.3 The positive solution to the Tarski problems -- 5.4 Tarski-like problems -- 6 Fully residually free groups I -- 6.1 Residually free and fully residually free groups -- 6.2 CSA groups and commutative transitivity -- 6.3 Universally free groups -- 6.4 Constructions of residually free groups -- 6.4.1 Exponential and free exponential groups -- 6.4.2 Fully residually free groups embedded in -- 6.4.3 A characterization in terms of ultrapowers -- 6.5 Structure of fully residually free groups -- 7 Fully residually free groups II -- 7.1 Fully residually free groups: limit groups -- 7.1.1 Geometric limit groups -- 7.2 JSJ-decompositions and automorphisms -- 7.2.1 Automorphisms of fully residually free groups -- 7.2.2 Tame automorphism groups.

7.2.3 The isomorphism problem for limit groups -- 7.2.4 Constructible limit groups -- 7.2.5 Factor sets and MR-diagrams -- 7.3 Faithful representations of limit groups -- 7.4 Infinite words and algorithmic theory -- 7.4.1 ℤn-free groups -- 8 Algebraic geometry over groups -- 8.1 Algebraic geometry -- 8.2 The category of G-groups -- 8.3 Domains and equationally Noetherian groups -- 8.3.1 Zero divisors and G-domains -- 8.3.2 Equationally Noetherian groups -- 8.3.3 Separation and discrimination -- 8.4 The affine geometry of G-groups -- 8.4.1 Algebraic sets and the Zariski topology -- 8.4.2 Ideals of algebraic sets -- 8.4.3 Morphisms of algebraic sets -- 8.4.5 Equivalence of the categories of affine algebraic sets and coordinate groups -- 8.4.6 The Zariski topology of equationally Noetherian groups -- 8.5 The theory of ideals -- 8.5.1 Maximal and prime ideals -- 8.5.2 Radicals and radical ideals -- 8.5.3 Some decomposition theorems for ideals -- 8.6 Coordinate groups -- 8.6.1 Coordinate groups of irreducible varieties -- 8.6.2 Decomposition theorems -- 8.7 The Nullstellensatz -- 9 The solution of the Tarski problems -- 9.1 The Tarski problems -- 9.2 Components of the solution -- 9.3 The Tarski-Vaught test and the overall strategy -- 9.4 Algebraic geometry and fully residually free groups -- 9.5 Quadratic equations and quasitriangular systems -- 9.6 Quantifier elimination and the elimination process -- 9.7 Proof of the elementary embedding -- 9.8 Proof of decidability -- 10 On elementary free groups and extensions -- 10.1 Elementary free groups -- 10.2 Surface groups and Magnus' theorem -- 10.3 Questions and something for nothing -- 10.4 Results on elementary free groups -- 10.4.1 Hyperbolicity and stable hyperbolicity -- 10.4.2 The retract theorem and Turner groups -- 10.4.3 Conjugacy separability of elementary free groups.

10.4.4 Tame automorphisms of elementary free groups -- 10.4.5 The isomorphism problem for elementary free groups -- 10.4.6 Faithful representations in PSL(2, C) -- 10.4.7 Elementary free groups and the Howson property -- 10.5 The Lyndon properties -- 10.5.1 The basic Lyndon properties -- 10.5.2 Lyndon properties in amalgams -- 10.5.3 The Lyndon properties and HNN constructions -- 10.5.4 The Lyndon properties in certain one-relator groups -- 10.5.5 The Lyndon properties and tree-free groups -- 10.6 The class of BX-groups -- 10.6.1 Big powers groups and univeral freeness -- 11 Discriminating and squarelike groups -- 11.1 Discriminating groups -- 11.2 Examples of discriminating groups -- 11.2.1 Abelian discriminating groups -- 11.2.2 Trivially discriminating groups and universal type groups -- 11.2.3 Nontrivially discriminating groups -- 11.3 Negative examples: nondiscriminating groups -- 11.3.1 Further negative examples in varieties -- 11.4 Squarelike groups and axiomatic properties -- 11.5 The axiomatic closure property -- 11.6 Further axiomatic information about discriminating and squarelike groups -- 11.7 Varietal discrimination -- 11.8 Co-discriminating groups and domains -- References -- Index -- De Gruyter Expositions in Mathematics.