The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup

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-Neumann-Neumann 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.

Locations

• Transactions of the American Mathematical Society - View - PDF