Central aspects of skew translation quadrangles, I

TL;DR

This paper advances the classification of finite skew translation generalized quadrangles, revealing new structural properties, solving key conjectures, and identifying symplectic quadrangles among odd order cases.

Contribution

It introduces a structure theory for root-elations, classifies skew translation quadrangles with distinct elation groups, and solves the Main Parameter Conjecture for a significant class.

Findings

Any skew translation quadrangle of odd order (s,s) is symplectic.

Classified skew translation quadrangles with distinct elation groups.

Developed a structure theory for root-elations in skew translation quadrangles.

Abstract

Except for the Hermitian buildings $\mathcal{H}(4,q^2)$, up to a combination of duality, translation duality or Payne integration, every known finite building of type $\mathbb{B}_2$ satisfies a set of general synthetic properties, usually put together in the term "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify finite skew translation generalized quadrangles. In the first installment of the series, as corollaries of the machinery we develop in the present paper, (a) we obtain the surprising result that any skew translation quadrangle of odd order $(s,s)$ is a symplectic quadrangle; (b) we determine all skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (c) we develop a structure theory for root-elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root-elations for each member, and hence all members are "central" (the main property needed to control STGQs, as which will be shown throughout); (d) we solve the Main Parameter Conjecture for a class of STGQs containing the class of the previous item, and which conjecturally coincides with the class of all STGQs.