Reversible and other generalised torsion elements in Seifert-fibered spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916596181630976 |
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| author | Das, Anushree Das, Debattam |
| author_facet | Das, Anushree Das, Debattam |
| contents | An element $a$ in a group $Γ$ is called \emph{reversible} if there exists $g \in Γ$ such that $gag^{-1}=a^{-1}$. The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on $3$ strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reversible and other generalised torsion elements in Seifert-fibered spaces Das, Anushree Das, Debattam Geometric Topology Primary 57K, Secondary 20H10, 20H05 An element $a$ in a group $Γ$ is called \emph{reversible} if there exists $g \in Γ$ such that $gag^{-1}=a^{-1}$. The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on $3$ strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular. |
| title | Reversible and other generalised torsion elements in Seifert-fibered spaces |
| topic | Geometric Topology Primary 57K, Secondary 20H10, 20H05 |
| url | https://arxiv.org/abs/2403.01541 |