The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3
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| Format: | Preprint |
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2025
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| _version_ | 1866912237595131904 |
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| author | Busch, Sira |
| author_facet | Busch, Sira |
| contents | We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $Δ$ of type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{H_m}$ for $n \in \mathbb{N}$ and $m \in \{3,4\}$. We show that we can extend these to the ambient building in the case that $Δ$ has type $\mathsf{B_n}$ or $\mathsf{C_n}$. With that, we obtain a different proof for the fact that buildings of type $\mathsf{B_n}$ and $\mathsf{C_n}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning $\mathsf{H_m}$, we will be able to see in a novel way why thick, spherical buildings of type $\mathsf{H_m}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $Δ$ of rank 3 have the Moufang property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3 Busch, Sira Group Theory 20E42 We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $Δ$ of type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{H_m}$ for $n \in \mathbb{N}$ and $m \in \{3,4\}$. We show that we can extend these to the ambient building in the case that $Δ$ has type $\mathsf{B_n}$ or $\mathsf{C_n}$. With that, we obtain a different proof for the fact that buildings of type $\mathsf{B_n}$ and $\mathsf{C_n}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning $\mathsf{H_m}$, we will be able to see in a novel way why thick, spherical buildings of type $\mathsf{H_m}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $Δ$ of rank 3 have the Moufang property. |
| title | The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3 |
| topic | Group Theory 20E42 |
| url | https://arxiv.org/abs/2502.12857 |