Colloquium de l'IDPLarge generic sets on spheres
Thursday 02 February 2012 14:00 - Orléans - Salle de Séminaire
A subset V of the 2-sphere is called 3-tame if every 3-point subset is contained in an open hemisphere. For example, 3 points chosen at random will satisfy the 3-tame condition with probability one. In fact, for any number n, a random choice of n points will, with probability one, be a 3-tame set. So there are many closed 3-tame sets. Every closed 3-tame set is contained in an open 3-tame set, and every open 3-tame set is contained in a maximal open 3-tame set. We'll explore the curious world of maximal open 3-tame sets using the following Lemma as a starting point: If V is an open (or closed) 3-tame set in the 2-sphere then there is at least one great circle which has empty intersection with V. We'll give two proofs of this lemma and also describe an important generalization. There are connections with the theory of modules over abelian groups and the Bieri-Neumann-Strebel invariant. The attached picture shows the simplest interesting example of a maximal open 3-tame subset of the 2-sphere: in this case there are exactly 7 great circles which do not meet the subset. In other examples we shall usually find that there are infinitely many such great circles.