Algebraic characterization of reversibility in the quaternionic Möbius group
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| Format: | Preprint |
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2024
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| author | Gongopadhyay, Krishnendu Lohan, Tejbir Mukherjee, Abhishek |
| author_facet | Gongopadhyay, Krishnendu Lohan, Tejbir Mukherjee, Abhishek |
| contents | An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group $\mathrm{PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that \[ [A]\in \mathrm{PSL}(2,\mathbb{H}) \text{ is reversible} \quad \Longleftrightarrow \quad β_A^{2}=δ_A^{2}, \] where $β_A$ and $δ_A$ are real conjugacy invariants associated with a lift $A\in \mathrm{SL}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in $\mathrm{SL}(2,\mathbb{H})$ and $\mathrm{PSL}(2,\mathbb{H})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic characterization of reversibility in the quaternionic Möbius group Gongopadhyay, Krishnendu Lohan, Tejbir Mukherjee, Abhishek Geometric Topology Group Theory Primary 51B10, 20E45, Secondary 15A21, 15B33 [2010] An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group $\mathrm{PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that \[ [A]\in \mathrm{PSL}(2,\mathbb{H}) \text{ is reversible} \quad \Longleftrightarrow \quad β_A^{2}=δ_A^{2}, \] where $β_A$ and $δ_A$ are real conjugacy invariants associated with a lift $A\in \mathrm{SL}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in $\mathrm{SL}(2,\mathbb{H})$ and $\mathrm{PSL}(2,\mathbb{H})$. |
| title | Algebraic characterization of reversibility in the quaternionic Möbius group |
| topic | Geometric Topology Group Theory Primary 51B10, 20E45, Secondary 15A21, 15B33 [2010] |
| url | https://arxiv.org/abs/2401.15374 |