Automorphisms and opposition in spherical buildings of exceptional type, IV: The $E_7$ case

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber onto an opposite chamber. This paper forms a significant part of a large project classifying domestic automorphisms of spherical buildings of exceptional type. In previous work the classifications for ${\mathsf{G}}_2$, ${\mathsf{F}}_4$ and ${\mathsf{E}}_6$ have been completed, and the present work provides the classification for buildings of type ${\mathsf{E}}_7$. In many respects this case is the richest amongst all exceptional types.