Groups with Tarski number 5

TL;DR

This paper constructs the first known example of a non-amenable group with Tarski number 5, expanding the understanding of paradoxical decompositions in group theory.

Contribution

It introduces a new group with Tarski number 5, filling a gap in the known possible Tarski numbers of groups.

Findings

Constructed a group with Tarski number 5

Established the existence of groups with Tarski number 5

Discussed related results for group actions

Abstract

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. Until now the only numbers which were known to be Tarski numbers of some groups were 4 and 6. We construct a group with Tarski number 5 and mention a related result for Tarski numbers of group actions.