Diophantine geometry over groups. III: Rigid and solid solutions.

DOI10.1007/BF02785359zbMath1133.20020MaRDI QIDQ2382241

Zlil Sela

Publication date: 28 September 2007

Published in: Israel Journal of Mathematics (Search for Journal in Brave)

zbMATH Keywords

quantifier eliminationfree groupsequations over groupselementary setspositive sentencesexceptional solutionsDiophantine geometry over groupsMakanin-Razborov diagrams: Diophantine setssolutions to systems of equationsparametric systems of equations

Mathematics Subject Classification ID

Noncommutative algebraic geometry (14A22) Applications of logic to group theory (20A15) Model-theoretic algebra (03C60) Geometric group theory (20F65) Decidability of theories and sets of sentences (03B25) Free nonabelian groups (20E05) Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) (20F10) Diophantine equations in many variables (11D72) Quasivarieties and varieties of groups (20E10) Basic properties of first-order languages and structures (03C07)

Related Items

The free group does not have the finite cover property, Limit groups for relatively hyperbolic groups. I: The basic tools., On CT and CSA groups and related ideas, Diophantine geometry over groups. V\(_1\): Quantifier elimination. I., Elementary theory of free non-abelian groups., The surface group conjecture: cyclically pinched and conjugacy pinched one-relator groups., On pro-\(p\) analogues of limit groups via extensions of centralizers., On Some Finiteness Properties in Infinite Groups, Diophantine geometry over groups. VIII: Stability., On Turner’s theorem and first-order theory, LIMIT GROUPS ARE CONJUGACY SEPARABLE, Subgroup properties of pro-\(p\) extensions of centralizers., Some lower bounds on Shelah rank in the free group, ALGEBRAIC LOGIC AND LOGICALLY-GEOMETRIC TYPES IN VARIETIES OF ALGEBRAS, Subgroup properties of fully residually free groups, A note on Lyndon properties in one-relator groups., Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence., A polynomial bound on solutions of quadratic equations in free groups., Classes of Groups Generalizing a Theorem of Benjamin Baumslag, A.D. Taimanov and model theory in Kazakhstan, Fields definable in the free group, Quadratic properties in group amalgams

Cites Work

• Unnamed Item
• Ergodic theory and free actions of groups on \({\mathbb{R}}\)-trees
• Acylindrical accessibility for groups
• Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence.
• Diophantine geometry over groups. II: Completions, closures and formal solutions
• Diophantine geometry over groups. I: Makanin-Razborov diagrams
• Stable actions of groups on real trees
• Counting maps from a surface to a graph.
• Bounding the complexity of simplicial group actions on trees
• DECIDABILITY OF THE UNIVERSAL AND POSITIVE THEORIES OF A FREE GROUP