A constructive way to compute the Tarski number of a group

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.

A survey of configurations

“…For Eq(g, E) satisfying the normal condition, there is a diagram that can give an upper bound for τ (G). The advantage of the second approach is to make the bound insofar as our choice of g and E permits (see [24]).…”

Countable-configurations and paradoxical decompositions

“…They used this concept particularly to characterize the amenability of groups [12]. Configurations are also applied to construct paradoxical decompositions and to estimate Tarski numbers of non-amenable groups [11,14,15,17]. This notion also was generalized to the case of semigroups [1].…”