Generalized polygon

In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959.

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.

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 bigger than 1) while the other infinite (these cases are called semi-finite).

Peter Cameron proved the non-existence of semi-finite generalized quadrangles with three points on each line, while Andries Brouwer and Bill Kantor independently proved the case of four points on each line.

The non-existence result for five points on each line was proved by G. Cherlin using Model Theory.

[1] No such results are known without making any further assumptions for generalized hexagons or octagons, even for the smallest case of three points on each line.

As noted before the incidence graphs of generalized polygons have important properties.

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