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.