Hyperdefinable groups in simple theories (Q2732509)

The author pursues his analysis of groups definable in simple theories. Here he studies hyperdefinable groups, i.e. groups \((G, \cdot)\) arising from a type-definable equivalence relation \(E\), a partial type \(\Gamma\) in the sort \(E\) and a type-definable \(E\)-invariant 3-ary relation \(\mu\) as \(G = \Gamma / E\) and \(\cdot = \mu / E\). NEWLINENEWLINENEWLINEFirst, the basic theory of these groups (generic types, chain conditions, locally connnected components, stabilizers) is developed, and internality and analyzability are examined. Then the paper deals with the existence of hyperdefinable quotient groups, and provides a version of the Weil-Hrushovski Group Chunk Theorem in this simple hyperdefinable setting. Locally modular groups are considered as well; it is shown that they are bounded-by-Abelian-by-bounded. Finally, hyperdefinable groups in supersimple theories are studied. Local versions of the \(SU\)-rank are analyzed, and a suitable arrangement of Zilber's Indecomposability Theorem is given.