The elementary theory of groups : a guide through the proofs of the Tarski conjectures /
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| Corporate Author: | |
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| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin ; Boston :
Walter de Gruyter GmbH & Co., KG,
[2014]
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| Series: | De Gruyter expositions in mathematics ;
60 |
| Subjects: | |
| Online Access: | Connect to this title online (unlimited simultaneous users allowed; 325 uses per year) |
Table of Contents:
- Machine generated contents note: 1. Group theory and logic: introduction
- 1.1. Group theory and logic
- 1.2. 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. 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. Tarski problems
- 5.1. Tarski problems
- 5.2. Initial work on the Tarski problems
- 5.3. 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 FZ[t]
- 6.4.3. 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. 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. 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. 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
- 8.4.4. Coordinate groups
- 8.4.5. Equivalence of the categories of affine algebraic sets and coordinate groups
- 8.4.6. Zariski topology of equationally Noetherian groups
- 8.5. 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. Nullstellensatz
- 9. solution of the Tarski problems
- 9.1. Tarski problems
- 9.2. Components of the solution
- 9.3. 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. 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. 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. Lyndon properties
- 10.5.1. basic Lyndon properties
- 10.5.2. Lyndon properties in amalgams
- 10.5.3. Lyndon properties and HNN constructions
- 10.5.4. Lyndon properties in certain one-relator groups
- 10.5.5. Lyndon properties and tree-free groups
- 10.6. 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. axiomatic closure property
- 11.6. Further axiomatic information
- 11.7. Varietal discrimination
- 11.8. Co-discriminating groups and domains.