# The Tarski numbers of groups

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 $G$ is the minimal number of pieces
in a paradoxical decomposition of $G$. 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 $2$-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 $t\ge 4$, the property
of having Tarski number $t$ is not invariant under quasi-isometry.
We also use $L^2$-Betti numbers to show that there exist groups with Tarski number $6$.
These provide the first examples of non-amenable groups without free subgroups
whose Tarski number has been computed precisely.