AS-configurations and skew-translation generalised quadrangles

Abstract

The only known skew-translation generalised quadrangles (STGQ) having order $(q,q)$, with $q$ even, are translation generalised quadrangles. Equivalently, the only known groups $G$ of order $q^3$, $q$ even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group $G$ admitting an AS-configuration, (ii) a classification of the STGQ of order $(8,8)$, and (iii) a classification of the STGQ of order $(q,q)$ for odd $q$ (using work of Ghinelli and Yoshiara).