Virtual retractions in free constructions

TL;DR

This paper investigates property (VRC) in fundamental groups of finite graphs of groups, providing criteria for when these groups have (VRC), and explores implications such as CAT(0) geometry and virtual specialness.

Contribution

It establishes a new criterion for property (VRC) in such groups using Euclidean geometry and linear algebra, and applies this to classify and analyze various group classes.

Findings

Groups with (VRC) are CAT(0)

Tubular groups with (VRC) are virtually free-by-cyclic

Criteria for (VRC) in graphs of finitely generated abelian groups

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.