Property (VRC) and virtual fibering for amalgamated free products

This paper focuses on studying properties of amalgamated free products $G=G_1*_{G_0} G_2$, where the amalgamated subgroup $G_0$ is virtually cyclic. First, we prove that if the factors $G_1$ and $G_2$ are finitely generated virtually abelian groups then $G$ can be mapped to another virtually abelian group so that this homomorphism is injective on each factor. We then present several applications of this result. In particular, we show that if $G_1$ and $G_2$ have property (VRC) (that is, every cyclic subgroup is a virtual retract), then the same is true for $G$. We also prove that $G$ inherits some residual properties (such as residual finiteness or virtual residual solvability) from the factors $G_i$, provided $G_0$ is a virtual retract of $G_i$, for $i=1,2$. Finally, we give necessary and sufficient conditions for $G$ to be (virtually) $F_m$-fibered. In particular, we fully characterize when an amalgamated product of two (finitely generated free)-by-cyclic groups over a cyclic subgroup is free-by-cyclic or virtually free-by-cyclic.