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AUTOMORPHISMS AND OPPOSITION IN TWIN BUILDINGS

Part of: Finite geometry and special incidence structures

Published online by Cambridge University Press:  08 March 2013

ALICE DEVILLERS ,
JAMES PARKINSON  and
HENDRIK VAN MALDEGHEM
ALICE DEVILLERS
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email alice.devillers@uwa.edu.au
JAMES PARKINSON*
Affiliation:
School of Mathematics & Statistics, The University of Sydney, NSW 2006, Australia
HENDRIK VAN MALDEGHEM
Affiliation:
Department of Mathematics, Ghent University, Krijgslaan 281, S22, 9000 Gent, Belgium email hvm@cage.UGent.be
*
✉jamesp@maths.usyd.edu.au
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Abstract

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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 (a key step in the proof is showing that every duality of a thick finite projective plane admits an absolute point). Our results also hold for all finite irreducible spherical buildings of rank at least 3, and imply that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

Keywords

twin buildingsprojective planesspherical buildings

MSC classification

Secondary: 51E24: Buildings and the geometry of diagrams
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 2 , April 2013 , pp. 189 - 201
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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