Appears in collection : 2014 - T1 - Random walks and asymptopic geometry of groups.

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.