Some questions related to free-by-cyclic groups and tubular groups

TL;DR

This work analyzes the interface between $CAT(0)$ geometry, free-by-cyclic structure, and tubular groups. It provides a precise $CAT(0)$ criterion for single-vertex tubular groups and demonstrates that, in this setting, $CAT(0)$-ness is equivalent to virtual specialness for free-by-cyclic tubular groups, with these groups in fact being $F_n$-by-$\mathbb{Z}$. It constructs counterexamples showing amalgamations along cyclic subgroups can fail to be (virtually) free-by-cyclic and that cyclic-subgroup separability does not imply property $(VRC)$, addressing several open questions in the negative. Finally, it clarifies distinctions between $RFRS$ and virtually $RFRS$, and discusses residual properties and embeddings, highlighting remaining questions at the intersection of geometric group theory and subgroup separability.

Abstract

We prove that a CAT(0) free-by-cyclic tubular group with one vertex is virtually special, but many of them cannot virtually act freely and cocompactly on CAT(0) cube complexes. This partially confirms a question of Brady--Soroko \cite[Section 9: Question 1]{BS} and answers a question of Lyman \cite[Question 1]{Ly} in the negative. Furthermore, we provide examples of free-by-cyclic groups amalgamated along cyclic subgroups that are not virtually free-by-cyclic. This answers negatively a question of Hagen--Wise \cite[Remark 3.6]{hw}. Lastly, we exhibit an example of a cyclic-subgroup-separable tubular group that does not have the property (VRC) (i.e. every cyclic subgroup is a virtual retract). This answers a question of Minasyan \cite[Question 11.6]{min} in the negative.

Figures (1)

• Figure 1: Immersed walls

Theorems & Definitions (84)

• Theorem 1.4
• Theorem 1.5: Corollary \ref{['cor1']} and Theorem \ref{['th7-1']}
• Remark 1.6
• Theorem 1.7: Theorem \ref{['thm:fr-cyc-vspel']}
• Theorem 1.9: \ref{['corlast']}
• Remark 1.10
• Theorem 1.12: Example \ref{['eg7.2']}
• Theorem 2.1
• Remark 2.2
• Lemma 2.3