2025/04/19 by Wu, Xiaolei, Ye, Shengkui · 1 citation
#20E06 #20E26 #20F65 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2504.14192
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.