Epimorphisms of Generalized Polygons Part 1: Geometrical Characterizations

A generalized polygon is a thick incidence geometry of rank 2 such that the girth of the incidence graph is twice the diameter of the incidence graph. These geometries are introduced by Tits [17] for group-theoretical purposes, but became an interesting research object in their own right. For an overview of the geometric study, see [19]. Obviously, the notion of morphism is an important one when dealing with geometries. For instance, monomorphisms are equivalent to embeddings of one geometry into the other (and on the group-theoretical level often give rise to maximal subgroups); isomorphisms clearly are needed to distinguish new geometries from old ones, but also to determine automorphism groups; epimorphisms can be used to construct quotient geometries or cover geometries (as the geometric counterpart of local fields). But in all these cases, the geometries considered in the literature are of the same kind, i.e., they have same gonality and diameter. In the present paper, we initiate the study of morphisms between generalized polygons of unequal gonality. We restrict ourselves to epimorphisms since we are motivated by some nice examples in this case. The study of monomorphisms and embeddings requires different techniques. To see the problem, a good starting point is Pasini’s theorem [13] that states that any epimorphism between two generalized n-gons, n > 2, is either an isomorphism or has infinite fibers. In particular, if an epimorphism is bijective if restricted to one point row, then it is a global isomorphism. This is no longer true for epimorphisms from a generalized m-gon to an n-gon, m 6= n, and a standard example is given is Section 3 below. It describes an epimorphism from the classical split Cayley hexagon over some field F to the ordinary Pappian projective plane over F with the property that line pencils and point rows are mapped bijectively onto line pencils and point rows, respectively. The paper is organized as follows. In section 2, we propose a quite general classification system for epimorphisms of geometries, from the local point of view. In the next section, Section 3, we give some examples, and we characterize geometrically our stan-