The Tarski number of a group action G O↓ 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 4 we construct a faithful transitive action of a free group with Tarski number κ. We also construct a group action G O↓ X with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of G are arbitrarily large.
- Paradoxical decomposition
- Stallings core
- Tarski number
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Discrete Mathematics and Combinatorics