Symmetry and Rigidity of Star-Shaped Coxeter Systems
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arXiv
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| Format: | Preprint |
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2026
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| author | Mahato, Arijit Naik, Tushar Kanta Patro, A Rameswar |
| author_facet | Mahato, Arijit Naik, Tushar Kanta Patro, A Rameswar |
| contents | We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the natural short exact sequence $1 \to \Inn (W) \to \Aut (W) \to \Out (W) \to 1$. Using Moussong's criteria for hyperbolicity, we show that these groups possess the $R_\infty$-property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetry and Rigidity of Star-Shaped Coxeter Systems Mahato, Arijit Naik, Tushar Kanta Patro, A Rameswar Group Theory Primary 20F55, Secondary 20F28, 20E36 We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the natural short exact sequence $1 \to \Inn (W) \to \Aut (W) \to \Out (W) \to 1$. Using Moussong's criteria for hyperbolicity, we show that these groups possess the $R_\infty$-property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems. |
| title | Symmetry and Rigidity of Star-Shaped Coxeter Systems |
| topic | Group Theory Primary 20F55, Secondary 20F28, 20E36 |
| url | https://arxiv.org/abs/2605.22065 |