Twin groups representations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917382969098240 |
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| author | Nasser, Mohamad N. |
| author_facet | Nasser, Mohamad N. |
| contents | We construct two representations of the twin group $T_n, n\geq 2$, namely $η_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $η_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $η_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $η_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twin groups representations Nasser, Mohamad N. Representation Theory Group Theory 20F36 We construct two representations of the twin group $T_n, n\geq 2$, namely $η_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $η_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $η_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $η_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions. |
| title | Twin groups representations |
| topic | Representation Theory Group Theory 20F36 |
| url | https://arxiv.org/abs/2507.15005 |