Tarski numbers of groups

Wed, 26/03/2014 - 10:30
The Tarski number of a group G is the minimal number of pieces in a paradoxical decomposition of it. 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 there are 2-generated groups with property (T) and arbitrarily large Tarski numbers.  
We also prove 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.
Joint work with Mikhail Ershov and Mark Sapir.