Constructing an Almost Hyperdefinable Group

Abstract

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

On almost orthogonality in simple theories

“…We can now apply tools from [BTW,Ben03]: LetP be the core-reduct of P . Then by [BTW], for every f ∈ P there is a group G f almost hyperdefinable over f , whose set of generic elements is the blow-up (alongf ) of P ; by [Ben03], there is a unique coreless polygroupP , almost hyperdefinable over ∅, such that P is the set of generic elements ofP .…”

“…In fact we shall construct a generic poly-chunk multi-acting on p in the sense of [Ben02]. This allows us to invoke the machinery of [BTW,Ben03,TW01] to obtain a coreless almost A-hyperdefinable polygroup, or, over some additional parameters, an almost hyperdefinable group acting transitively on an almost hyperdefinable set X, whose generic elements are interbounded (over independent parameters) with realizations of p. If the theory was stable to start with, the group obtained will be the original binding group.…”

On the fine structure of the polygroup blow-up

“…This paper continues [BTW01], seeking a better understanding of two fundamental concepts introduced therein: polygroups and the blow-up procedure.…”

“…In the discussions that led to the writing of [BTW01], one of the main sources of intuition was the double coset polygroup G/ /H = {HaH : a ∈ G}. But then, this could be considered as a problem on its own: can any polygroup be written in this form?…”

“…We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G/ /H, and a polygroup chunk theorem.…”