Non-abelian representations of the slim dense near hexagons on 81 and 243 points

Abstract

Let S =(P,L) be a partial linear space with point set P and line set L. We suppose that S is slim, i.e., that every line of S is incident with precisely three points. For distinct points x,y ∈ P, we write x ∼ y if they are collinear. In that case, we denote by xy the unique line containing x and y and define x ∗y by xy ={x,y,x∗y}.For x ∈P, we define x⊥ :={x}∪{y ∈P :y ∼x}.Ifx,y∈P, then d(x,y)denotes the distance between x and y in the collinearity graph of S.