## Generalized hexagons containing a subhexagon

I have recently uploaded a joint paper with Bart, “On generalized hexagons of order $(3, t)$ and $(4, t)$ containing a subhexagon”,on arXiv and submitted it for publication. In this work we extend the results of my first paper, which I discussed here, by proving the following:

Let $q \in \{2, 3, 4\}$ and let $S$ be a generalized hexagon isomorphic to the split Cayley hexagon $\mathrm{H}(q)$ or its dual $\mathrm{H}(q)^D$. Then the following holds for any generalized hexagon $S'$ that contains $S$ as a full subgeometry: (1) $S'$ is finite; (2) if $q \in \{2, 4\}$ and $\mathcal{H} \cong \mathrm{H}(q)$, then $S' = S$.

This result is what one might expect in general since (1) we do not know of any semi-finite generalized polygons and (2) we do know of any generalized hexagons which properly contain $\mathrm{H}(q)$ as a full subgeometry, for $q$ a power of a prime $p \neq 3$ (when $q$ is a power of $3$, $\mathrm{H}(q)$ is isomorphic to its dual, which is always contained in a generalized hexagon of order $(q, q^3)$). But since we are nowhere near classifying finite generalized polygons, this problem seems to be pretty hard in general. Also, existence/non-existence of semi-finite generalized polygons is a major open problem in incidence geometry which has only been resolved for specific cases of generalized quadrangles (when every line has $s \in \{3, 4, 5\}$ points on it). So for now we satisfy ourselves with this small contribution.

There is a nice counting based lemma in our paper which I really like and I hope that it can be used to obtain more results for generalized hexagons containing subhexagons. Let’s see what the lemma says.

Let $S$ be a generalized hexagon of order $(s, t)$ contained in a generalized hexagon $S'$ as a full subgeometry (every line of $S$ is also a full line of $S'$). Then using standard arguments for generalized polygons, we first show that every line of $S'$ must have precisely $s + 1$ points on it. It directly follows from the axioms of a generalized hexagon that every point of $S'$ is at distance at most $2$ from $S$. So, we have three “types” of points in $S'$, those contained in $S$, those at distance $1$ from $S$ and those at distance $2$ from $S$. These three types of points can also be characterised by the kind of substructures of $S$ they induce when we take all points of $S$ at non-maximal distance from a given point of $S'$. We call these substructures singular, semi-singular and ovoidal hyperplanes, respectively (there is a good reason why these substructures of $S$ are called “hyperplanes” of $S$; see Section 3 of the paper for exact definitions).

Now if $S$ is a proper subgeometry of $S'$, then it can be shown that there must be a line $L$ of $S'$ which does not contain any point of $S$. Say this line contains $n_L$ points of the third type. Then our lemma says that for any two points $x, y$ on $L$, the substructures of $S$ induced by $x$ and $y$ intersect each other in precisely $s + 1 - n_L$ points. The lemma is proved using a result of Bart (Proposition 4.7 in Polygonal Valuations), which we reprove in the paper without relying too much on the theory of polygonal valuations, followed by simple counting.

How do we use this lemma? Well, if for a given generalized hexagon $S$ one can show that every pair of semi-singular hyperplanes in $S$ intersect each other in more than $s + 1$ points, then there cannot be any generalized hexagon $S'$ be which contains $S$ as a proper full subgeometry as otherwise we can derive a contradiction by finding an appropriate line of $S'$ which does not contain any points of $S$. This is what we check  for the case when $S$ is isomorphic to $\mathrm{H}(2)$ or $\mathrm{H}(4)$ and thus obtain part (2) of our main result. Our check mostly consists of some clever computations in a computer model of these geometries as we do not really understand how semi-singular (or ovoidal) hyperplanes of split Cayley hexagons and their duals behave in general.

To prove part (1) of our main result we show that if every point of $S'$ is at distance at most $1$ from $S$, or equivalently, there are no points of the third type in $S'$, then through every point of $S'$ there are only finitely many lines. Therefore, if we show that a given generalized hexagon $S$ does not contain any ovoidal hyperplanes a.k.a. $1$-ovoids a.k.a. distance-$2$ ovoids, then from this result we get that every generalized hexagon containing $S$ as a full subgeometry is finite. These $1$-ovoids are simply sets of points which intersect every line in a unique point, thus in graph theoretical terms these are the exact hitting sets of the hypergraph obtained from the generalized hexagon by identifying each line by the set of points it contains. Now when $S$ is isomorphic to $\mathrm{H}(2)^D$ (a geometry that has $63$ points and $63$ lines) it is easy to show via an exhaustive computer search or via an old result of Frohardt and Johnson which classifies all hyperplanes of this geometry, that it does not contain any $1$-ovoids. But in general this problem is quite hard. For $\mathrm{H}(4)^D$ we use the non-existence result proved by me and Ferdinand in a recent paper.

The last remaining case of our main result is when $S$ is isomorphic to $H(3)$. The extra complication here is that $H(3)$ is isomorphic to its dual, and hence it is contained in the dual twisted triality hexagon of order $(3, 27)$. Moreover, it has $1$-ovoids. What we do here is again use the lemma on intersection sizes, but this time we show that for every point $x$, every semi-singular hyperplane  with center $x$ (again, see Section 3 of the paper for the definition) of $S$ intersects every ovoidal hyperplane containing $x$ in more than $3$ points. From this we show that there are no type 3 points in any generalized hexagon $S'$ which contains $S$ has a full subgeometry.

Sadly, we have reached our computational limitations at $q = 4$ and it seems like for larger cases we will need more ideas and a better understanding of these three types of hyperplanes in split Cayley hexagons and their duals. If in general one can prove that every semi-singular hyperplane of $\mathrm{H}(q)$  (or $\mathrm{H}(q)^D)$ through a fixed point $x$  intersects every ovoidal hyperplane containing $x$ (whenever such a hyperplane exists) in more than $q$ points, then this would imply that there is no semi-finite generalized hexagon with $q + 1$ points on each line, containing $\mathrm{H}(q)$ (or $\mathrm{H}(q)^D)$ as a subgeometry. That would be really nice.

There are some natural open problems arising from this work which we list in the last section of our paper. In particular, I would be really happy to see a proof of the following conjecture in the near future.

Every generalized hexagon containing the generalized hexagon of order $(2, 1)$, which arises from the incidence graph of the Fano plane, is finite.

This is the smallest case where none of our techniques work.