Steven Buechler

Geometrical stability theory is a powerful set of model-theoretic tools that can lead to structural results on models of a simple first-order theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M equipped with a class of finitary relations R is strongly λ−homogeneous if orbits under automorphisms of (M, R) have finite character in the following sense: Given α an ordinal < λ ≤ |M | and sequences ¯ a = { a b in) have the same orbit, for all n and i 1 < · · · < in < α, then f (¯ a) = ¯ b for some automorphism f of (M, R). In this paper strongly λ−homogeneous models (M, R) in which the elements of R induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called " dividing " , agrees with forking independence when (M, R) is saturated. The concept central to the development of geometrical stability theory for saturated structures, namely the canonical base, is also shown to exist in this setting. These results broaden the scope of the methods of geometrical stability theory. This paper attempts to give a self-contained development of dividing theory (also called forking theory) in a strongly homogeneous structure. Dividing is a combinatorial property on the invariant relations on a structure that have yielded deep results for the models of so-called " simple " first-order theories. Below we describe for the nonspecialist how this paper fits in the broader context of geometrical stability theory. Naturally, some background in first-order model theory helps to understand these motivating results, however virtually no knowledge of logic is assumed in this paper. Readers desiring a more thorough description of geometrical stability theory are referred to the surveys [Hru97] and [Hru98]. Traditionally, geometrical stability theory is a collection of results that apply to definable relations on arbitrary models of a complete first-order theory. It is equivalent and convenient to restrict our attention to the definable relations on a fixed representative model of the theory, called a …