Generalized Quadrangles Of Order $(s, s^2)$. IV. Translations, Moufang And Fong-Seitz

TL;DR

This paper classifies finite translation generalized quadrangles of order (s, s^2) with specific properties, linking them to Moufang quadrangles and the Fong-Seitz theorem on BN-pairs, advancing understanding of their structure.

Contribution

It provides a classification of certain finite translation generalized quadrangles of order (s, s^2) with a kernel of size at least 3 and a regular line not incident with the translation point.

Findings

Classified all such translation generalized quadrangles with the given properties.

Established connections to Moufang quadrangles.

Applied results to the Fong-Seitz theorem on BN-pairs.

Abstract

In the period 1994-1999 Thas wrote a series of three papers on generalized quadrangles of order $(s, s^2)$. In this Part IV we classify all finite translation generalized quadrangles of order $(s, s^2)$ having a kernel of size at least 3, containing a regular line not incident with the translation point. There are several applications on generalized quadrangles of order $(s, s^2)$ having at least 2 translation points, on Moufang quadrangles, and concerning the theorem of Fong and Seitz classifying all groups with a BN-pair of rank 2.