Some questions related to free-by-cyclic groups and tubular groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910915097526272 |
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| author | Wu, Xiaolei Ye, Shengkui |
| author_facet | Wu, Xiaolei Ye, Shengkui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some questions related to free-by-cyclic groups and tubular groups Wu, Xiaolei Ye, Shengkui Group Theory Metric Geometry 20E26, 20E06, 20F65 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. |
| title | Some questions related to free-by-cyclic groups and tubular groups |
| topic | Group Theory Metric Geometry 20E26, 20E06, 20F65 |
| url | https://arxiv.org/abs/2504.14192 |