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arXiv · 1401.2202

The Tarski numbers of groups

Mikhail Ershov·Gili Golan·Mark Sapir
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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 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.

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Mikhail Ershov, Gili Golan, Mark Sapir. 2014-01-09. The Tarski numbers of groups. https://doi.org/10.1016/j.aim.2015.07.010

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