Symmetry and Rigidity of Star-Shaped Coxeter Systems

TL;DR

This paper characterizes the automorphism groups of star-shaped Coxeter systems, explores their structural properties, and solves the isomorphism problem for this class.

Contribution

It provides a complete description of automorphisms, investigates the $R_ fty$-property, and establishes rigidity results for star-shaped Coxeter groups.

Findings

Automorphism groups decompose into inner, diagram, transvections, and partial conjugations.

Groups have the $R_ fty$-property under Moussong's hyperbolicity criteria.

The paper solves the isomorphism problem for star-shaped Coxeter systems.

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.