Introduction

Stallings [6] showed that a group G has more than one end if and only if G ≍ A*FB, where F is finite, A ≠ F ≠ B, or G is an HNN-extension with finite edge group F.

A finitely generated group G is said to be accessible if it is the fundamental group of a graph of groups in which all edge groups are finite and every vertex group has at most one end. We say that G is inaccessible if it is not accessible.

Let d(G) denote the minimal number of generators of the finitely generated group G. It follows from Grushko's Theorem that d(G*H) = d(G) + d(H). It follows that G is a free product of indecomposable groups, i.e. groups which cannot be written as a non-trivial free product. The problem of accessibility is whether we can replace the free product with free product with finite amalgamation in the last statement. (The number of HNN-decompositions is bounded by d(G).) However, there is no analogue of Grushko's Theorem. In fact, if G is accessible then any process of sucessively decomposing G, and the factors that arise in the process, terminates after a finite number of steps. See [2] for a proof of this and related results.

Linnell [5] proved that if G is finitely generated then, for any reduced decomposition of G as a graph of groups X in which all edge groups are finite, there is a bound B such that where E is the edge set of X.