The Tarski numbers of groups

Mikhail Ershov, Gili Golan, Mark Sapir

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Pages (from-to)21-53
Number of pages33
JournalAdvances in Mathematics
StatePublished - 2 Oct 2015
Externally publishedYes


  • Amenability
  • Cost
  • Golod-Shafarevich groups
  • L<sup>2</sup>-Betti number
  • Paradoxical decomposition
  • Tarski number

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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