Property (VRC) and virtual fibering for amalgamated free products

TL;DR

This paper investigates properties of amalgamated free products with virtually cyclic subgroups, establishing conditions for virtual properties, residual properties, and virtual fibering, with applications to free-by-cyclic groups.

Contribution

It provides new criteria for virtual properties and fibering in amalgamated free products over virtually cyclic subgroups, including characterizations for free-by-cyclic structures.

Findings

Amalgamated free products can be mapped to virtually abelian groups preserving factors.

Property (VRC) is inherited under certain conditions.

Residual properties are preserved when the amalgamated subgroup is a virtual retract.

Abstract

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.