No abstract

Abstract.We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G.

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.