4:00 pm Thursday, February 6, 2014

Mark Sapir (Vanderbilt)

It is well known (the Hausdorff-Banach-Tarski paradox), that a sphere of radius 1 can be decomposed into 5 pieces which can be then rearranged using rotations of the sphere to obtain two spheres of radii 1. The reason for this paradox is that the group of rotations of the sphere contains a free non-Abelian subgroup, and a free non-Abelian group has a paradoxical decomposition with 4 pieces, i.e., its Tarski number is 4. Von Neumann defined the class of amenable groups as groups which do not have paradoxical decompositions.

For a non-amenable group, its Tarski number shows how close the group is to being amenable (it is related to other invariants of non-amenable groups such as cogrowth, spectral radius and the isoperimetric constant). For example, having a free non-amenable subgroup is equivalent to having Tarski number 4. For quite some time, though, very little was known about possible Tarski numbers of groups. We show that the set of possible Tarski numbers is infinite, and in fact for every sufficiently large n, there is a Tarski number between n and 2n. We also study how the Tarski numbers behave under group theoretic constructions, and find an example of a group with Tarski number 6, the first number >4 which is proved to be the Tarski number of a group. This is a joint work with Mikhail Ershov and Gili Golan.