Generalized Polygons

Definitions

Some progress on locally-primitive generalized polygons

Let $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathbf{I})$ be a finite geometry. The incidence graph $\Gamma$ of $\mathcal{S}$ is the graph with the vertex set $\mathcal{P}\cup \mathcal{L}$ and flags of $S$ as edges.

$\mathcal{S}$ is a finite generalized $n$-gon if $\Gamma$ is a connected graph of diameter $n$ and girth $2n$.

A finite generalized $n$-gon $\mathcal{S}$ is called thick (firm) if every vertex of $\Gamma$ has degree at least $3$ ($2$).

There are five equivalent definitions of generalized polygon, which is mentioned here. For example, the type (IV) is as follows.

from here

$\mathcal{S}$ is a finite generalized $n$-gon if the following two axioms are satisfied:

• $\mathcal{S}$ contains no ordinary $k$-gon as a sub-geometry, for $2⩽k<n$.
• Any two elements $x,y\in \mathcal{P}\cup \mathcal{L}$ are contained in some ordinary $n$-gon as a sub-geometry in $\mathcal{S}$.

Definition

An automorphism of $\mathcal{S}$ is a permutation $\theta$ preserving $\mathcal{P}$, $\mathcal{L}$ and incidence relation $\mathrm{I}$. The full automorphism group $\mathrm{Aut}(\mathcal{S})$ is the group of all automorphisms of $\mathcal{S}$.

Restriction of $n$

Feit-Higman, 1964

Finite thick generalized $n$-gons exist only for $n\in \{3,4,6,8\}$.

When $n=3$, $\mathcal{S}$ is a projective plane.

When $n=4$, $\mathcal{S}$ is a generalized quadrangle.

When $n=6$, $\mathcal{S}$ is a generalized hexagon.

When $n=8$, $\mathcal{S}$ is a generalized octagon.

Order $(s,t)$

Some progress on locally-primitive generalized polygons

Let $\mathcal{S}$ be a finite generalized $n$-gon. Then $\mathcal{S}$ has order $(s,t)$ if $\Gamma$ is bi-regular of degree $(s+1,t+1)$. Remark that $s$ and $t$ are also defined here.

V Maldeghem, Generalized Polygons

Every thick generalized $n$-gon has an order $(s,t)$ with $s,t⩾2$; if $n=3$, then $s=t$, and if $n=8$ then $s\ne t$.

Examples

Easy Examples

The following slide comes from here.

Well-known Examples

Furthermore, more well-known examples are displayed as follows:

• Thick GQs: Examples of Thick GQs
• Thick GHs and GOs: Examples of Thick GHs and GOs

Certain Symmetry

Integral Conditions

Definition

$\mathcal{S}$ is flag-transitive (antiflag-transitive), if it has an automorphism group $G⩽\mathrm{Aut}(\mathcal{S})$ such that $G$ acts transitively on all flags (antiflags) of $\mathcal{S}$.

$\mathcal{S}$ is point-primitive (line-primitive), if it has an automorphism group $G⩽\mathrm{Aut}(\mathcal{S})$ acting primitively on $\mathcal{P}$ ($\mathcal{L}$).

Remark. All known GHs and GOs are flag-transitive, point-primitive, and line-primitive; all known classical GQs are flag-transitive, point-primitive, and line primitive. All of them are mentioned here.

There are two open problems:

• Classify all flag-transitive finite generalized polygons? With conjecture:
  • $n=4$, Kantor 1991, classical or finite exceptions.
  • $n=6,8$, are the known examples.
• Classify all point-primitive (line-primitive) finite generalized polygons?

Local Conditions

Definition

Let $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathrm{I})$ be a generalized $n$-gon and $G⩽\mathrm{Aut}(\mathcal{S})$. $\mathcal{S}$ is called $G$-locally $\mathrm{P}$, if for each vertex $u$ in $\Gamma$, the stabilizer $G_u$ has the property $\mathrm{P}$ in $\Gamma (u)$, where $\Gamma (u)$ is the neighbor of $u$ in $\Gamma$.

Remark. dwd gives some relations between local and integral conditions, see here.

Parameter Conditions

See here.

Some Useful Tools

See here:

• Benson type argument
• Fixed element structure
• Normal quotient graphs