If a finitely generated torsion free group K has the property that all
finitely generated subgroups S of K are either small or have growth constant
bounded uniformly away from 1 then a non proper HNN extension G of K, that is a
semidirect product of K by the integers, has the same property. Here small
means cyclic or, if the automorphism has no periodic conjugacy classes, free
abelian of bounded rank.