Girth alternative for HNN extensions

Abstract The girth of a graph is the length of a shortest cycle in it. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For a finitely generated group, we can define its girth as the supremum of girths of Cayley graphs of it with respect to all finite generating sets. A given class of finitely generated groups is said to satisfy Girth Alternative if any group from this class is either virtually solvable or has infinite girth. We prove the Girth Alternative for a sub-class of HNN extensions as well as for a sub-class of amalgamated free products of finitely generated groups, and indicate counterexamples to show that beyond our class, the alternative fails in general. We also prove the Girth Alternative for HNN extensions of non-elementary word hyperbolic groups.