Generalized polygon

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In combinatorial theory, a generalized polygon is an incidence structure introduced by Jacques Tits. Generalized polygons 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 polygon is also a near polygon.

Definition[edit]

A generalized 2-gon (or a digon) is a partial linear space where each point is incident to each line. For n > 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 < m < n.
  • It has 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.

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[edit]

  • 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[edit]

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 t1/2st2.
  • If n = 6, then st is a square, and t1/3st3.
  • If n = 8, then 2st is a square, and t1/2st2.
  • 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[edit]

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.[1] Several classes of extremal expander graphs are obtained from generalized polygons.[2]

See also[edit]

References[edit]