The paper “The Tarski numbers of groups” by Mikhail Ershov from Virginia, Gili Golan from Bar Ilan and myself can be found here.
Here is an abstract:
The Tarski number of a non-amenable group is the minimal number of pieces
in a paradoxical decomposition of . In this paper we investigate how
Tarski numbers may change under various group-theoretic operations. Using these
estimates and the theory of Golod-Shafarevich groups, we show that
the Tarski numbers of -generated non-amenable groups can be arbitrarily large, and that the Tarski numbers of finite index subgroups of a given finitely generated non-amenable group can be arbitrarily large. In particular, for some number , the property
of having Tarski number is not invariant under quasi-isometry.
We also use -Betti numbers to show that there exist groups with Tarski number .
These provide the first examples of non-amenable groups without free subgroups
whose Tarski number has been computed precisely.