On the Howson Property of HNN-Extensions with Nilpotent Base Group and Amalgamated Free Products of Nilpotent Groups

Abstract

We give necessary and sufficient conditions under which an amalgamated free product of finitely generated nilpotent groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). Also we prove that if G = ⟨ t, K | t −1 At = B ⟩, where K is a finitely generated and infinite nilpotent group and A, B non-trivial infinite proper subgroups of K, then G is not a Howson group. The problem of deciding when an ascending HNN-extension of a finitely generated nilpotent group is a Howson group is still open.