Domesticity in Generalized Quadrangles

Abstract

An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4.