Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917991997767680 |
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| author | Parkinson, James Van Maldeghem, Hendrik |
| author_facet | Parkinson, James Van Maldeghem, Hendrik |
| contents | An automorphism of a spherical building is called \textit{domestic} if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types $\mathsf{F}_4$, $\mathsf{E}_6$, and $\mathsf{E}_7$ have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least~$3$ by classifying such automorphisms in the $\mathsf{E}_8$ case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of $B$-cosets, with $B$ the stabiliser of a fixed choice of chamber. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case Parkinson, James Van Maldeghem, Hendrik Group Theory An automorphism of a spherical building is called \textit{domestic} if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types $\mathsf{F}_4$, $\mathsf{E}_6$, and $\mathsf{E}_7$ have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least~$3$ by classifying such automorphisms in the $\mathsf{E}_8$ case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of $B$-cosets, with $B$ the stabiliser of a fixed choice of chamber. |
| title | Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.14184 |