Tarski numbers of group actions

TL;DR

This paper constructs specific group actions demonstrating prescribed Tarski numbers, revealing new insights into paradoxical decompositions and their behavior under subgroup restrictions.

Contribution

It introduces methods to construct faithful transitive group actions with exact Tarski numbers and explores their behavior under finite index subgroup restrictions.

Findings

Constructed a faithful transitive action of a free group with Tarski number k for any k>3.

Developed a group action of a free group with Tarski number 6 where restrictions have arbitrarily large Tarski numbers.

Provided new techniques for analyzing paradoxical decompositions in group actions.

Abstract

The Tarski number of an action of a group G on a set X is the minimal number of pieces in a paradoxical decomposition of it. For any k>3 we construct a faithful transitive action of a free group of rank k-1 with Tarski number k. Using similar techniques we construct a group action of a free group F with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of F are arbitrarily large.