The group configuration theorem states that given a certain forking configuration, one can construct from it a canonical group action. Following the development of the theory of many-valued function germs by Itay Ben-Yaacov (Group configurations and germs in simple theories, JSL 67(4):1581-1600, 2002), the group was found by Ben-Yaacov, Tomasic and Wagner Constructing an almost hyperdefinable group; however it is living in the collection of almost hyperimaginaries (classes modulo invariant equivalence relations covered uniformly by boundedly many hyperdefinable sets). The group action was then recovered by Tomasic and Wagner Applications of the group configuration theorem in simple theories.
The binding group theorem says that if a type p is almost orthogonal to qw but non-orthogonal to q, then the restriction to (the realizations of) p of the group of automorphisms (of the monster model) fixing q (and all parameters) is definable. In the simple context this group has been studied by Bradd Hart and Ziv Shami; however it may not properly reflect forking (adding a random bipartite graph between p and q will trivialize it). One may alternatively consider the group of elementary permutations of p over q (which is still trivialized by an added bipartite random graph, but not quite as bad), see Shami and Wagner. Using the group configuration theorem (and consequently almost hyperimaginaries), the group corresponding to the forking geometry was found by Ben-Yaacov and Wagner On almost orthogonality in simple theories.
A survey of these results has appeared in the Bulletin of Symbolic Logic.
|Frank O. Wagner
Institut Camille Jordan
Institut universitaire de France