HYPERDEFINABLE GROUPS IN SIMPLE THEORIES

We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by-bounded. Finally, we analyze hyperdefinable groups in supersimple theories.

Relations, homogeneity and group quotients

A set with a relation is isomorphic to a group quotient under the condition described as weak homogeneity, and to the quotient of a group with relation preserved by right and left translations if the homogeneity is strengthened. A method of constructing these group quotients and, furthermore, all such very homogeneous spaces, is described and an illustrative example given.

The Nakayama Map and Ramification for Maximally Complete Fields

Let K be a maximally complete valued field and let L be a totally ramified Galois extension of K with Galois group G. Assume (i) the value group quotient of L|K is cyclic and (ii) there exists an unramified cyclic extension of K of the same degree as L. Then there is an isomorphism of Ga onto a subgroup A/N(L×) of K×/N(L×) which maps the ramification group Gi onto AiN(L×)/N(L×) for all i > 0 where Ai = {x ∊ A|v(x ‒ 1) ≧ i}. This generalizes certain results of Local Class Field Theory.