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 $U$ 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 $N$ is a f.g. normal subgroup of $G$ not contained in $U$, then $NU$ has finite index in $G$.