Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2107.09985 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911786822795264 |
|---|---|
| author | Hillman, J. A. |
| author_facet | Hillman, J. A. |
| contents | If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $β_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $β_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then $G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_09985 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Nilpotent groups with balanced presentations. II Hillman, J. A. Geometric Topology 20F18, 20J05, 57N13 If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $β_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $β_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then $G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$. |
| title | Nilpotent groups with balanced presentations. II |
| topic | Geometric Topology 20F18, 20J05, 57N13 |
| url | https://arxiv.org/abs/2107.09985 |