Finitely Presented Extensions by Free Groups

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitely presented groups with free quotients. Our main objective in this note is to prove the following Theorem 1. Let G be a finitely presented group with a free quotient of rank n. Then G is an HNN extension with n stable letters, of a finitely generated group B and finitely generated associated subgroups. It is worth noting that if N is the normal closure in G of B, then N is the fundamental group π(Γ) of a graph Γ of groups, where Γ is the Cayley graph of a free group of rank n. The vertex groups are isomorphic to B and the edge groups are isomorphic to the associated subgroups involved in the decomposition of G as an HNN–extension and are therefore all finitely generated. There are a number of consequences of Theorem 1. The first of these is the following theorem of Bieri and Strebel [3] which is a special case of Theorem 1. Corollary 1. A finitely presented group with an infinite cyclic quotient is an HNN extension of a finitely generated group with a single stable letter and finitely generated associated subgroups. In order to formulate the second consequence of Theorem 1, we recall that a group is termed coherent if all of its finitely generated subgroups Date: November 26, 2006. 1991 Mathematics Subject Classification. Primary 20.