Peter Kropholler: Generic open subsets of spheres and applications to the Bieri--Neumann--Strebel invariant

A subset of a sphere is called m-tame if every m-point subset is contained in an open hemisphere. This property first arose in the Bieri--Groves conjecture concerning which metabelian groups have property FP_m. In joint work with Bieri and Owens, we study the property and discover some interesting combinatorial aspects as we develop the theory of maximal open n-tame subsets of an n-1 sphere.