Affine Subdivision, Steerable Semigroups, and Sphere Coverings

Let ∆ be a Euclidean n-simplex and let {∆j} denote a finite union of simplices which partition ∆. We assume that the partition is invariant under the affine symmetry group of ∆. A classical example of such a partition is the one obtained from barycentric subdivision, but there are plenty of other possibilities. (See §4.1, or else [Sp, p. 123], for a definition of barycentric subdivision.) Our partition gives rise to an affine subdivision rule for ∆, which may be iterated. To subdivide each ∆j , we choose an affine map Aj with ∆j = Aj(∆), and then partition ∆j into the collection {Aj(∆i)}. The affine invariance of the partition translates into the fact that our partition of ∆j is independent of the (n+ 1)! different choices for Aj . Now we iterate. A basic question one can ask is Does the iteration of the subdivision rule produce a dense set of shapes of simplices? By shape of a simplex, we mean a simplex considered mod similarities. In [BBC] this question was raised and answered affirmatively for the case of 2-dimensional barycentric subdivision. In [S] we got the same result in 3 dimensions. In general, a first step for the kind of density results just mentioned is as follows: Let Cn be the collection of all n-dimensional simplices obtained by iteratively applying the subdivision rule. Let Ωn be the collection of matrices of the form ± L | det(L)|1/n ,