Mark Sapir - The Tarski numbers of groups.

Mark Sapir (Vanderbilt University, USA)
The Tarski number of a non-amenable group is the minimal
number of pieces in a paradoxical decomposition of the group. It is
known that a group has Tarski number 4 if and only if it contains a
free non-cyclic subgroup, and the Tarski numbers of torsion groups are
at least 6. It was not known whether the set of Tarski numbers is
infinite and whether any particular number greater than 4 is the Tarski number of
a group. We prove that the set of possible Tarski numbers is infinite
even for 2-generated groups with property (T), show that 6 is the
Tarski number of a group (in fact of any group with large enough first L_2-Betti number), and prove several results showing how the Tarski number behaves under extensions of groups. This is a joint work with Mikhail Ershov and Gili Golan.