Generalized polygon

In combinatorial theory, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized n-gon with n even is also a near polygon.

Definition

A generalized 2-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line.

For $n \geq 3$ a generalized n-gon is an incidence structure ($P,L,I$), where $P$ is the set of points, $L$ is the set of lines and $I\subseteq P\times L$ is the incidence relation, such that:

• It is a partial linear space.
• It has no ordinary m-gons as subgeometry for $2 \leq m < n$.
• It has an ordinary n-gon as a subgeometry.
• For any $\{A_1, A_2\} \subseteq P \cup L$ there exists a subgeometry ($P', L', I'$) isomorphic to an ordinary n-gon such that $\{A_1, A_2\} \subseteq P' \cup L'$.

An equivalent but sometimes simpler way to express these conditions is: consider the bipartite incidence graph with the vertex set $P \cup L$ and the edges connecting the incident pairs of points and lines.

• The girth of the incidence graph is twice the diameter n of the incidence graph.

From this it should be clear that the incidence graphs of generalized polygons are Moore graphs.

A generalized polygon is of order (s,t) if:

• all vertices of the incidence graph corresponding to the elements of $L$ have the same degree s + 1 for some natural number s; in other words, every line contains exactly s + 1 points,
• all vertices of the incidence graph corresponding to the elements of $P$ have the same degree t + 1 for some natural number t; in other words, every point lies on exactly t + 1 lines.

We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.

The dual of a generalized n-gon ($P,L,I$), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the inverse relation of $I$. It can easily be shown that this is again a generalized n-gon.

Examples

• The incidence graph of a generalized digon is a complete bipartite graph K_s+1,t+1.

• For any natural n ≥ 3, consider the boundary of the ordinary polygon with n sides. Declare the vertices of the polygon to be the points and the sides to be the lines, with set inclusion as the incidence relation. This results in a generalized n-gon with s = t = 1.

• For each group of Lie type G of rank 2 there is an associated generalized n-gon X with n equal to 3, 4, 6 or 8 such that G acts transitively on the set of flags of X. In the finite case, for n=6, one obtains the Split Cayley hexagon of order (q, q) for G₂(q) and the twisted triality hexagon of order (q³, q) for ³D₄(q³), and for n=8, one obtains the Ree-Tits octagon of order (q, q²) for ²F₄(q) with q = 2²ⁿ⁺¹. Up to duality, these are the only known thick finite generalized hexagons or octagons.

Feit-Higman theorem

Walter Feit and Graham Higman proved that finite generalized n-gons with s ≥ 2, t ≥ 2 can exist only for the following values of n:

2, 3, 4, 6 or 8.

Moreover,

• If n = 2, the structure is a complete bipartite graph.
• If n = 3, the structure is a finite projective plane, and s = t.
• If n = 4, the structure is a finite generalized quadrangle, and t^1/2 ≤ s ≤ t².
• If n = 6, then st is a square, and t^1/3 ≤ s ≤ t³.
• If n = 8, then 2st is a square, and t^1/2 ≤ s ≤ t².
• If s or t is allowed to be 1 and the structure is not the ordinary n-gon then besides the values of n already listed, only n = 12 may be possible.

If s and t are both infinite then generalized polygons exist for each n greater or equal to 2. It is unknown whether or not there exist generalized polygons with one of the parameters finite and the other infinite (these cases are called semi-finite).

Applications

As noted before the incidence graphs of generalized polygons have important properties. For example, every generalized n-gon of order (s,s) is a (s,2n) cage. They are also related to expander graphs as they have nice expansion properties. Several classes of extremal expander graphs are obtained from generalized polygons.

See also

• Building (mathematics)
• (B, N) pair
• Ree group
• Moufang plane
• Near polygon

References

• .
• .
• .
• .
• .