Generalized quadrangles with two translation points
by
Koen Thas
Ghent University, Ghent, Belgium

Suppose S is a finite generalized quadrangle (GQ) of order (s, t), s, t > 1, and suppose that L is a line of S. A symmetry about L is an automorphism of the GQ which fixes every line of S meeting L (including L). A line is called an axis of symmetry if there is a full group of symmetries of size s about this line.
Suppose L and M are non-concurrent axes of symmetry of the GQ S; then S is called a span-symmetric generalized quadrangle (SPGQ) with base-span sp(L, M). It was a longstanding conjecture that every SPGQ of order s > 1 is classical, i.e. isomorphic to the GQ Q(4, s) which arises from a nonsingular parabolic quadric in PG(4, s), a result we proved recently using the classification of the finite split BN-pairs of rank 1 (by E. E. Shult and C. Hering, W. M. Kantor and G. M. Seitz) and universal central extensions of groups.
The general problem of classifying SPGQ's of order (s, t), s, t > 1 and s =/= t, seems hopeless at present, although it is worthwhile mentioning that W. M. Kantor was able to show that in such a case we necessarily have that t = s².
As a first step in the aforementioned classification, we focus on a special class of SPGQ's of order (s, s²), s > 1. Let us first recall that a point of a generalized quadrangle is a translation point if every line incident with it is an axis of symmetry, and a GQ with a translation point is often called a translation generalized quadrangle.
In the present talk, we will discuss the classification of generalized quadrangles which have at least two distinct translation points.

Date received: March 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-18.