This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem.

By Stalling's Theorem a group with more than one end splits over a finite subgroup, i.e. can be written as an HNN-Extension or a free product with amalgamation (over a finite subgroup).

However when I was working through the proofs of Dunwoody, Dunwoody & Krön and Krön, which all use Bass-Serre Theory, I was wondering if there is any way to detect if we are dealing with an HNN-Extension or a free product with amalgamation. Primarily I was thinking of some property of the Cayley graph or the action on it by the group in question. For example one could consider the induced action on the end space of the Cayley graph.

In my opinion this is a natural question and I would be grateful for any comment or references regarding this.

EDIT: I am looking for properties of the group, its Cayley graph, the action of the group on its Cayley graph and/or its end space etc., which help to distinguish if the group, supposed to split by Stalling's Theorem, is indeed an HNN-Extension *or* a free product with amalgamation.

For some statements one probably has to make additional assumptions such as the group is finitely generated or finitely presented etc.

ora free product with amalgamation. Maybe I should be more precise in the original post. $\endgroup$ – M.U. Aug 19 '15 at 11:12