Recent Results on Subquadrangles of Generalized Quadrangles.
Koen Thas (Universiteit Gent)
A subquadrangle (subGQ) of a generalized quadrangle (GQ) is a substructure of the GQ which is a GQ itself by the induced incidence. Let Q(5,q) be the GQ which consists of the points and lines of a nonsingular elliptic quadric in PG(5,q) (the projective 5-space over the finite field with q elements). Then every two distinct non-concurrent lines are contained in q + 1 distinct Q(4,q)-subGQs, where Q(4,q) is the GQ consisting of the points and lines of a nonsingular parabolic quadric in PG(4,q).
In the last couple of years, several authors (e.g., M. R. Brown and M. Lavrauw, J. A. Thas and K. Thas, K. Thas) have gained new insights in the theory of subGQs, by classifying certain types of GQs (such as ``elation GQs'', ``translation GQs'', and ``flock GQs'') which have certain types of subGQs (such as Q(4,q)-subGQs or subGQs in some special position). Many new characterizations of Q(5,q) were obtained in this way.
In the present talk, I want to elaborate on these results.