The Tarski numbers of groups

TL;DR

This paper explores how Tarski numbers, which measure paradoxical decompositions of non-amenable groups, can vary under group operations, providing new examples with specific Tarski numbers and analyzing their properties.

Contribution

It demonstrates that Tarski numbers of 2-generated non-amenable groups can be arbitrarily large and constructs groups with Tarski numbers 5 and 6, the first known examples without free subgroups.

Findings

Tarski numbers can be arbitrarily large for 2-generated non-amenable groups.

Existence of groups with Tarski numbers 5 and 6 without free subgroups.

New bounds and properties of Tarski numbers under group operations.

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.