Configuration of groups and paradoxical decompositions

A constructive way to compute the Tarski number of a group

“…Remark 3.5. In [7] the authors constructed the paradoxical decomposition under the paradoxical condition. That is, a subsystem of Eq(g, E) is equivalent to BX = 0,…”

“…Otherwise the decomposition is not complete but by a process described in the proof of [7,Proposition 1.2] it can be changed into a complete one. Proof.…”

“…Remark 3.5. In [7] the authors constructed the paradoxical decomposition under the paradoxical condition. That is, a subsystem of Eq(g, E) is equivalent to BX = 0, where each row of B has nonnegative entries and is of the form m i=1 (L ji i − L ki i ), for some j i , k i ∈ {0, .…”

“…Therefore it is valuable to construct the paradoxical decomposition for such a group, using configurations. This problem which was originally asked by Willis is answered partially in [7]. In that paper the paradoxical decomposition was constructed under a paradoxical condition.…”

A survey of configurations

“…We omit the proofs but for the sake of importance, we point out that the proofs are strongly based on the following two lemmas. For details see [12].…”