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arXiv · 2605.22065

Symmetry and Rigidity of Star-Shaped Coxeter Systems

Arijit Mahato·Tushar Kanta Naik·A Rameswar Patro
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Abstract

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.

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Arijit Mahato, Tushar Kanta Naik, A Rameswar Patro. 2026-05-21. Symmetry and Rigidity of Star-Shaped Coxeter Systems. https://arxiv.org/abs/2605.22065

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