Configuration equivalence is not equivalent to isomorphism

Abstract

Giving a condition for the amenability of groups, Rosenblatt and Willis first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will prove this conjecture, due to Rosenblatt and Willis, whether the configuration equivalent groups include the free non-Abelian group of the same rank or not. We show that two-sided equivalent groups have same class numbers.