TITLE: finite index subgroups of amalgamated free products
QUESTION [7 upvotes]: Let $G=A\ast_C B$ be an amalgamated free product of two infinite groups such that $A,B$ do not have any finite index normal proper subgroups. Is it true that $G$ does not have any finite index normal proper subgroups?

REPLY [7 votes]: Yes.
Note that a group $G$ has a proper finite-index normal subgroup if and only if there exists a nontrivial homomorphism from $G$ to some finite group.
Consider any homomorphism
$$\varphi\colon A * B \to F,$$
where $F$ is a finite group.  By hypothesis, $\varphi(A)$ and $\varphi(B)$ must be trivial.  Since $A$ and $B$ generate the domain, it follows that $\varphi$ is trivial.  Thus $A* B$ cannot have any proper finite-index normal subgroups.  Since $A*_C B$ is a quotient of $A*B$, the same must hold for $A*_C B$.