Finite graphs of groups with isomorphic fundamental groups

Abstract

In this paper the answer is given to the following question: How are the finite graphs of groups, the fundamental groups of which are isomorphic, while the edge and vertex groups satisfy Restrictions 1 and 2 cited below, connected? i) The vertex groups are ~ -groups, that is, nonrepresentable in the form of a nontrivial free product with union or a free extension. 2) The edge groups are co-Hopf, that is not isomorphic to their proper subgroups. An attempt to answer this question under those same conditions was made in [4], however, the proof of the basic theorem of this paper contains mistakes, and its conclusion is refuted by the counterexamples constructed below. One can, it is true, note that under a stronger condition than 1 the requirement that the vertex groups are Abelian the theorem from [4] is true. As a corollary of the basic theorem, we obtain the decidability of the isomorphism problem for the fundamental groups of finite graphs of finite groups, proved previously in [5].