Abstract

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 $F_4$, $E_6$, and $E_7$ 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 $E_8$ 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.

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Mathematical Subject Classification

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