Quadratic properties in group amalgams
-
Benjamin Fine
,
Aila Rosenberger
Abstract
Lyndon in [Combinatorial Group Theory, Springer-Verlag, 1977] proved that in a free group F each solution set {x, y, z} of the quadratic equation x2y2z2 = 1 generates a cyclic group. This theorem launched the whole theory of equations over free groups which led eventually to the solution of the Tarski conjectures. Since Lyndon's theorem can be phrased as a universal sentence, it is true in all fully residually free groups, if we replace cyclic by abelian. In the present paper we consider amalgams of groups which satisfy Lyndon's theorem. We prove that this property and several other related properties involving quadratic and quadratic-type equations are preserved under free products with malnormal amalgamated subgroups. The situation in HNN groups is more complicated.
© de Gruyter 2011
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- A proof that Thompson's groups have infinitely many relative ends
- Quadratic properties in group amalgams
- Stabilizers of ℝ-trees with free isometric actions of FN
- On factorizations of finite groups with ℱ-hypercentral intersections of the factors
- On the involution module of SL2(2ƒ)
- A note on element centralizers in finite Coxeter groups
- Lattice-defined classes of finite groups with modular Sylow subgroups
- On groups with all proper subgroups of finite exponent
- A case-free characterization of hyperbolic Coxeter systems
- Corrigendum: Some class size conditions implying solvability of finite groups
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Articles in the same Issue
- A proof that Thompson's groups have infinitely many relative ends
- Quadratic properties in group amalgams
- Stabilizers of ℝ-trees with free isometric actions of FN
- On factorizations of finite groups with ℱ-hypercentral intersections of the factors
- On the involution module of SL2(2ƒ)
- A note on element centralizers in finite Coxeter groups
- Lattice-defined classes of finite groups with modular Sylow subgroups
- On groups with all proper subgroups of finite exponent
- A case-free characterization of hyperbolic Coxeter systems
- Corrigendum: Some class size conditions implying solvability of finite groups
- On the q-tensor square of a group
