A constructive way to compute the Tarski number of a group

TL;DR

This paper introduces a constructive method using configurations and matrix properties to compute the Tarski number of non-amenable groups, providing bounds based on path counting in associated diagrams.

Contribution

It presents a novel constructive approach to determine the Tarski number using configurations and combinatorial matrix properties.

Findings

Constructed paradoxical decompositions using configurations.

Provided an upper bound for the Tarski number via diagram path counting.

Established a new method for analyzing non-amenable groups.

Abstract

The Tarski number of a group $G$ is the minimal number of the pieces of paradoxical decompositions of that group. Using configurations along with a matrix combinatorial property we construct paradoxical decompositions. We also compute an upper bound for the Tarski number of a given non-amenable group by counting the number of paths in a diagram associated to the group.