Symmetry and Rigidity of Star-Shaped Coxeter Systems

We provide a complete description of the automorphism group $\text{Aut}(W)$ 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 $1\to \text{Inn}(W)\to \text{Aut}(W)\to \text{Out}(W)\to 1$. Using Moussong’s criteria for hyperbolicity, we show that these groups possess the $R_{\mathrm{\infty }}$-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.