Property (VRC) and virtual fibering for amalgamated free products

TL;DR

This work advances the understanding of amalgamated free products $G=G_1*_{G_0} G_2$ with a virtually cyclic $G_0$ by embedding $G$ into a finitely generated virtually abelian group $E$ in a way that preserves the factor injections. The authors develop a robust toolkit—functorial splittings into wreath-product ambient groups, and a detailed analysis via Bass–Serre theory and BNSR invariants—to prove that (i) amalgams preserve property (VRC) and residual properties when $G_0$ is a virtual retract of the factors, and (ii) provide necessary and sufficient criteria for (virtual) $F_m$-fibering of such amalgams, including a full characterization for when an amalgam of two finitely generated free-by-cyclic groups over a cyclic subgroup is (virtually) free-by-cyclic. The results yield concrete fibering criteria that apply beyond virtually RFRS, expanding the landscape of virtual fibering in graph-of-groups settings and offering tools for constructing or obstructing fibering in complex combinations. Overall, the paper supplies a versatile framework for transferring finiteness, separability, and fibering properties through amalgamations, with implications for the structure and geometry of the resulting groups.

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.

Theorems & Definitions (127)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Corollary 1.5
• Definition 1.8
• Proposition 1.10
• Corollary 1.11
• proof