Narrow Results

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.

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math.48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if $G$ and $H$ have same two-sided configuration sets and $N$ is a normal subgroup of $G$ with polycyclic or FC quotient, then there is a normal subgroup $𝔑$ of $H$ such that $G/N\cong H/𝔑$. Also, we show that if $G$ and $H$ are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.