On generalized hexagons of order $(3, t)$ and $(4, t)$ containing a subhexagon

TL;DR

This paper proves the non-existence of semi-finite generalized hexagons of certain orders containing known hexagons as subgeometries for q=3 and 4, and shows the split Cayley hexagon of order 4 cannot be a subgeometry.

Contribution

It establishes new non-existence results for semi-finite generalized hexagons of orders (3,t) and (4,t), extending previous work and strengthening known containment results.

Findings

No semi-finite generalized hexagons of order (3,t) or (4,t) contain known hexagons as full subgeometries.

The split Cayley hexagon of order 4 cannot be properly contained as a full subgeometry in any generalized hexagon.

Alternative proof provided for the case q=2 previously studied by the authors.

Abstract

We prove that there are no semi-finite generalized hexagons with $q + 1$ points on each line containing the known generalized hexagons of order $q$ as full subgeometries when $q$ is equal to $3$ or $4$, thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when $q$ is equal to $2$ was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order $4$ we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.