Dependence relations on homogeneous groups and homogeneous expansions of Hilbert spaces

A notion of independence called non-forking was introduced by Shelah to study problems related to classification theory. Forking has been a very useful tool in first-order theories for analyzing the structure of models. For example, this analysis has lead to characterizing models in a classifiable theory through a collection of invariants.; Buechler and Lessmann defined a notion of independence, called freeness, working inside a large strongly homogeneous structure. When this notion satisfies the expected properties, the structure is called simple. When a simple structure has a bound on the size of independent extensions of types over sets, the structure is called simple stable.; Working in a simple stable strongly homogeneous model, we use tools from first-order logic to study groups. We define the notions of generics, stabilizers and connected components. We generalize results from first-order stability theory and show that connected 1-based groups are abelian and that the geometry of an SU-rank one 1-based group is that of a module over a ring. We define internality and we prove the existence of a strongly homogeneous group. When M is |M|-strongly homogeneous, the group we get is the binding group.; We study freeness in strongly homogeneous expansions of Hilbert spaces. We prove the following theorems:; Theorem 0.1. Let (H, +, ⟨, ⟩, {lcub}E_λ{rcub}) be a strongly δ-homogeneous δ-saturated Hilbert space where {lcub}E_λ{rcub} is the resolution of the identity of a self-adjoint operator T, δ > ($2\aleph 0$)+. This structure is $\aleph 1$-simple stable and for f ∈ H, A ⊂ H, A = BDD( A), tp(f/A) is stationary and the projection of f over A can be identified with the canonical base of f over A.; Theorem 0.2. Let X be a set with | X| > $2\aleph 0$.{09}Let l2 (X) be the set of square integrable functions from X to $R$ and + and × their pointwise addition and multiplication. Then (l2(X), +, ⟨, ⟩, ×) is $\aleph 1$-simple stable. Furthermore, the structure is 1-based and for f ∈ H, A = acl( A) a closed subspace, tp(f/A) is stationary and the projection of f over A can be taken to be the canonical base of f over A.