Automorphisms and opposition in sphericalbuildings of exceptional type, V : The E8 case

An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types F4, E6, and E7 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 E8 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.