AS-configurations and skew-translation generalised quadrangles

The only known skew-translation generalised quadrangles (STGQ) having order (q,q)(q,q), with q even, are translation generalised quadrangles. Equivalently, the only known groups G   of order q3q3, 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)(8,8), and (iii) a classification of the STGQ of order (q,q)(q,q) for odd q (using work of Ghinelli and Yoshiara).