In this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangle C induced from a group configuration, we build a canonical hyperdefinable homogeneous space equivalent to C. When C is principal, we can choose the homogeneous space principal as well.

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...) on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)

Abstract.1. We show that ifpis a real type which is internal in a set Σ of partial types in a simple theory, then there is a typep′ interbounded withp, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.2. Ifpis a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts onpwhile fixing Σ generically In casepis Σ-internal andTis stable, this is (...) the binding group ofpover Σ. (shrink)

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