Orders On Free Metabelian Groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913456537468928 |
|---|---|
| author | Wang, Wenhao |
| author_facet | Wang, Wenhao |
| contents | A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-order of non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14630 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Orders On Free Metabelian Groups Wang, Wenhao Group Theory Logic A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-order of non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language. |
| title | Orders On Free Metabelian Groups |
| topic | Group Theory Logic |
| url | https://arxiv.org/abs/2210.14630 |