Virtual retractions in free constructions

TL;DR

This paper analyzes the property $VRC$ for fundamental groups of finite graphs of groups, with a focus on vertex groups that are finitely generated virtually abelian. It proves a pivotal criterion: $G$ has $VRC$ iff there exists a homomorphism to a Euclidean-by-finite group $P=\mathbb{R}^n \rtimes Q$ that is injective on every vertex group, enabling $VRC$ verification through linear/affine geometry. The authors develop edge-approximating frameworks and nearly linear independence criteria to extend $VRC$ to wide classes, including trees and general graphs of abelian groups, and demonstrate deep geometric consequences: $VRC$ implies CAT$(0)$ actions and, for tubular groups, virtual specialness. They also explore stability properties of $VRC$ and its relation to $LR$, providing both positive results and counterexamples that delineate the boundaries of these virtual retract notions. Overall, the work links algebraic properties of group decompositions to geometric actions, yielding practical criteria for $VRC$-ness and broad applications to CAT$(0)$ geometry and subgroup separability.

Abstract

A group $G$ has property (VRC) if every cyclic subgroup is a virtual retract. This property is stable under many standard group-theoretic constructions and is enjoyed by all virtually special groups (in the sense of Haglund and Wise). In this paper we study property (VRC) for fundamental groups of finite graphs of groups. Our main criterion shows that the fundamental group of a finite graph of finitely generated virtually abelian groups has (VRC) if and only if it has a homomorphism to a Euclidean-by-finite group that is injective on all vertex groups. This result allows us to determine property (VRC) for such groups using basic tools from Euclidean Geometry and Linear Algebra. We use it to produce examples and to give sufficient criteria for fundamental groups of finite graphs of finitely generated abelian groups with cyclic edge groups to have (VRC). In the last two sections and in the appendix we give applications of property (VRC). We show that if a fundamental group of a finite graph of groups with finitely generated virtually abelian vertex groups has (VRC) then it is CAT($0$). We also show that tubular groups with (VRC) are virtually free-by-cyclic and virtually special.

Figures (2)

• Figure 1: Graph of groups for the group $L$ in \ref{['RAAG not Lerf']}
• Figure 2: The graph of groups $(\mathcal{G},\Gamma)$ in \ref{['ex: graph of abelian n.l.i']}

Theorems & Definitions (139)

• Corollary 1.1
• Example 1.2
• Corollary 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.8
• Proposition 1.9
• Proposition 1.10
• Corollary 1.11