Diophantine Geometry over Groups IX: Envelopes and Imaginaries

Abstract

This paper is the ninth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets in free and hyperbolic groups. In the ninth paper we associate a Diophantine set with a definable set, and view it as the Diophantine envelope of the definable set. We use the envelope and duo limit groups that were used in proving stability of the theory of free and torsion-free hyperbolic groups [Se9], to study definable equivalence relations, and in particular, to classify imaginaries in these groups.

References in corpus (1)

• On groups and fields interpretable in torsion-free hyperbolic groups

Cited by in corpus (8)

• Diophantine Geometry over Groups X: The Elementary Theory of Free Products of Groups
• On groups and fields interpretable in torsion-free hyperbolic groups
• Ampleness in the free group
• Free and Hyperbolic Groups are not Equational
• Algebraic & definable closure in free groups
• A note on ampleness in the theory of non abelian free groups
• On ampleness and pseudo-Anosov homeomorphisms in the free group
• Fields interpretable in the free group