Mark Sapir (Centennial Professor, Vanderbilt University, and George A. Miller Visiting Professor, Univ. of Illinois) Tarski numbers Abstract: It is known since Hausdorff, Banach and Tarski that one can decompose a 2sphere into 4 pieces, move the pieces using rotations of the sphere, and obtain two spheres of the same radius (assuming the Axiom of Choice). Thus a sphere with the group of rotations acting on it has a paradoxical decomposition with 4 pieces. In general if we have a group G acting on a set X, then the Tarski number of X is the minimal number of pieces in a paradoxical decomposition of X. For example, if G acts on itself by left multiplication, then we can talk about the Tarski number of G. I will show how to use GolodShafarevich groups to prove that the set of possible Tarski numbers of groups is infinite. I will also show how to use l_2Betti numbers of groups and cost of group actions to construct groups with Tarski numbers 5 and 6. Note that 4, 5 and 6 are the only numbers that are currently known to be Tarski numbers of groups. This is a joint work with Gili Golan and Mikhail Ershov. 
