Tarski Number and Configuration Equations

The concept of configuration of groups which is defined in terms of finite partitions and finite strings of elements of the group is presented by Rosenblatt and Willis. To each set of configurations, a finite system of equations known as configuration equations, is associated. Rosenblatt and Willis proved that a discrete group G is amenable if and only if every possible instance of its configuration equations admits a normalized solution. In this paper we compare the existence of such solutions for different systems. We prove that if a system of configuration equations has no normalized solution, then every system related to a refinement of the initial partition, has no normalized solution, as well. The Tarski number of a non-amenable group is the smallest number of the pieces of its paradoxical decompositions. In the present paper we also provide a relation between the Tarski numbers of the subgroups of two configuration equivalent groups.