Groups, Geometry, and Dynamics

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Volume 10, Issue 3, 2016, pp. 933–950
DOI: 10.4171/GGD/371

Published online: 2016-09-16

Tarski numbers of group actions

Gili Golan[1]

(1) Bar-Ilan University, Ramat Gan, Israel

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.

Keywords: Tarski number, paradoxical decomposition, amenability, Stallings core

Golan Gili: Tarski numbers of group actions. Groups Geom. Dyn. 10 (2016), 933-950. doi: 10.4171/GGD/371