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arXiv · 1203.6172

Automorphisms and opposition in twin buildings

Alice Devillers·James Parkinson·Hendrik Van Maldeghem
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Abstract

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite. The main ingredient of the proof is a lemma that states that every duality of a thick finite projective plane admits an absolute point, i.e., a point mapped onto an incident line. Our results also hold for all finite irreducible spherical buildings of rank at least 3, and as a consequence we deduce that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

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Alice Devillers, James Parkinson, Hendrik Van Maldeghem. 2012-03-28. Automorphisms and opposition in twin buildings. https://arxiv.org/abs/1203.6172

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