Minimal Cubings

We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev’s original construction. We also show that almost invariant sets can be chosen to be in very good position. Let G be a finitely generated group, and let H1, . . . , Hn be subgroups. For i = 1, . . . , n, let Xi be a nontrivial Hi–almost invariant subset of G. In [6], Sageev gave a natural construction of a cubing C(X1, . . . , Xn) with a G–action which reflects the way in which the translates of the Xi’s intersect each other. In order to give the reader a feel for this, we start by discussing a simple and closely related topological example. For other simple examples, the reader ∗Partially supported by NSF grants DMS 034681 and 9626537