The Tarski numbers of groups

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 known properties of Golod-Shafarevich groups, we show that the Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We also use the cost of group actions to show that there exist groups with Tarski numbers 5 and 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely.
26 pages. v4: We have added the result about the existence of groups with Tarski number 5 which previously appeared in a separate paper of the second author arXiv:1406.2097. Section 5 has been rewritten. Minor changes in Appendix B. We also replaced left group actions by right group actions