Let $G=A\ast_C B$ be a non-trivial free product with amalgamation. Then, if $C$ has index greater than two in $A$ or $B$, $G$ has infinitely many ends if it is infinite and the amalgamating subgroup is finite, by Stallings' Theorem on Ends of Groups. My question is,
What does it mean for the amalgamating subgroup to be finite?
Do you have to take two finite subgroups of $A$ and $B$, say $H$ and $K$ respectively, and amalgamate them (i.e. they are finite before the amalgamation), or can $H$ and $K$ be infinite but they map to a finite subgroup of $G$ (so they are finite after the amalgamation). For example,
$G=F_2 \ast_{C_n} C_n=\langle a, b, c; b=c, c^n\rangle$
and here $H=\langle b\rangle\cong\mathbb{Z}$ with $K=\langle c\rangle\cong C_n$. (I'm pretty sure this is a free product with amalgamation - but I'm never quite sure about this stuff...)