Classification of skew translation generalized quadrangles, I

We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B-2), more precisely in the (large) subtheory of skew translation generalized quadrangles ("STGQs"). Some of these involve, and solve, long-standing open problems

Characterisations of elementary pseudo-caps and good eggs

In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $\mathrm{PG}(4n-1, q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(\mathcal{O})$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw

Singer quadrangles

[no abstract available

Some characterizations of the exceptional planar embedding of W(2)

AbstractIn this paper, we study the representation of W(2) in PG(2,4) related to a hyperoval. We provide a group-theoretic characterization and some geometric ones

Covers of generalized quadrangles

We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci. 126 (2016), 591-612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from theta-covers, and consider related problems

A geometric proof of a theorem on antiregularity of generalized quadrangles

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201