## Opposition diagrams for automorphisms of large spherical buildings

### J. Parkinson and H. Van Maldeghem

#### Abstract

Let $$\theta$$ be an automorphism of a thick irreducible spherical building $$\Delta$$ of rank at least $$3$$ with no Fano plane residues. We prove that if there exist both type $$J_1$$ and $$J_2$$ simplices of $$\Delta$$ mapped onto opposite simplices by $$\theta$$, then there exists a type $$J_1\cup J_2$$ simplex of $$\Delta$$ mapped onto an opposite simplex by $$\theta$$. This property is called "cappedness". We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano plane residues. In these buildings automorphisms are not necessarily capped.

Keywords: Buildings, projective spaces, polar spaces, domestic automorphism.

: Primary 20E42; secondary 51E24.

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 Monday, December 18, 2017