The Elementary Theory of Groups
A Guide through the Proofs of the Tarski Conjectures
1st Edition
Published on 29. October 2014
XIV, 307 pages
978-3-11-034203-1 (ISBN)
Description
After being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. Both proofs involve long and complicated applications of algebraic geometry over free groups as well as an extension of methods to solve equations in free groups originally developed by Razborov. This book is an examination of the material on the general elementary theory of groups that is necessary to begin to understand the proofs. This material includes a complete exposition of the theory of fully residually free groups or limit groups as well a complete description of the algebraic geometry of free groups. Also included are introductory material on combinatorial and geometric group theory and first-order logic. There is then a short outline of the proof of the Tarski conjectures in the manner of Kharlampovich and Myasnikov.
More details
Other editions
Additional editions
Benjamin Fine | Anthony Gaglione | Alexei Myasnikov
The Elementary Theory of Groups
A Guide through the Proofs of the Tarski Conjectures
E-Book
10/2014
1st Edition
De Gruyter
€210.00
Available for download
Benjamin Fine | Anthony Gaglione | Alexei Myasnikov
The Elementary Theory of Groups
A Guide through the Proofs of the Tarski Conjectures
Book
10/2014
1st Edition
De Gruyter
€229.00
Article exhausted; check different version
Benjamin Fine | Anthony Gaglione | Alexei Myasnikov
The Elementary Theory of Groups
A Guide through the Proofs of the Tarski Conjectures
Book
09/2014
1st Edition
De Gruyter
€210.00
Shipment within 7-9 days
Persons
B. Fine, Fairfield U.; A. Gaglione, US Naval Academy; A. Myasnikov, McGill U.; G. Rosenberger, U. of Hamburg; D. Spellman, Temple U.
Content
- Intro
- Preface
- 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 FZ[t]
- 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 ??-domains
- 8.3.2 Equationally Noetherian groups
- 8.3.3 Separation and discrimination
- 8.4 The affine geometry of Z-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 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
- 11.7 Varietal discrimination
- 11.8 Co-discriminating groups and domains
- References
- Index
