Solution to an open problem on extremal generalized hexagons

A finite generalized 2d-gon of order (s, t) with d ∈ {2, 3, 4} and s 6= 1 is called extremal if t attains its maximal possible value sed , where e2 = e4 = 2 and e3 = 3. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized 2d-gon of order (s, t) to be extremal has so far only be solved for quadrangles and octagons. In this paper, we obtain a solution for the generalized hexagons, hereby settling an open problem that essentially dates back from the end of the 70’s, see [8, 12, 13]. In fact, we obtain a solution as a particular case of a more general result on extremal regular near hexagons.