4 Rich groups, weak second-order logic, and applications
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Abstract
In this chapter we initiate a study of first-order rich groups, i. e., groups where the first-order logic has the same power as the weak second-order logic. Surprisingly, there are quite a few finitely generated rich groups, they are somewhere inbetween hyperbolic and nilpotent groups (these are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev’s problems in various groups.
Abstract
In this chapter we initiate a study of first-order rich groups, i. e., groups where the first-order logic has the same power as the weak second-order logic. Surprisingly, there are quite a few finitely generated rich groups, they are somewhere inbetween hyperbolic and nilpotent groups (these are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev’s problems in various groups.
Chapters in this book
- Frontmatter I
- Introduction V
- Contents IX
- 1 Model theory and groups 1
- 2 Independence and interpretable structures in nonabelian free groups 51
- 3 Quantifier elimination algorithm to boolean combination of ∃∀-formulas in the theory of a free group 87
- 4 Rich groups, weak second-order logic, and applications 127
- 5 Rigid solvable groups. Algebraic geometry and model theory 193
- Index 231
Chapters in this book
- Frontmatter I
- Introduction V
- Contents IX
- 1 Model theory and groups 1
- 2 Independence and interpretable structures in nonabelian free groups 51
- 3 Quantifier elimination algorithm to boolean combination of ∃∀-formulas in the theory of a free group 87
- 4 Rich groups, weak second-order logic, and applications 127
- 5 Rigid solvable groups. Algebraic geometry and model theory 193
- Index 231
