Tarski numbers of group actions

Abstract

The Tarski number of a group action $G\curvearrowright X$ is the minimal number of pieces in a paradoxical decomposition of it. In this paper we solve the problem of describing the set of Tarski numbers of group actions. Namely, for any $k\ge 4$ we construct a faithful transitive action of a free group with Tarski number $k$. We also construct a group action $G\curvearrowright X$ with Tarski number $6$ such that the Tarski numbers of restrictions of this action to finite index subgroups of $G$ are arbitrarily large.