Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated free product w.r.t the 'pieces' of the space) are well-known theorems. However I often read that they are kind of analogues or that Stalling's Theorem is the 'combinatorial' version or something like that.
I was wondering if this could be made a bit more precise?
I know that there are some major differences. Indeed Stalling's Theorem deals with amalgamated free products and HNN-Extensions over finite subgroups, where the Seifert-van Kampen Theorem 'just' deals with amalgamated free products.
In this sense I was also wondering if there is some sort of 'Seifert-van Kampen Theorem' for HNN-Extensions?