Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types $\mathsf{F}_4$, $\mathsf{E}_6$, and $\mathsf{E}_7$ have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least~$3$ by classifying such automorphisms in the $\mathsf{E}_8$ case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of $B$-cosets, with $B$ the stabiliser of a fixed choice of chamber.