Of finite geometries of type $C₃$ with thick lines

Abstract

Several facts suggest the conjecture that every finite geometry of type $C₃$ with thick lines is either a bulding or flat ( see [4] , [5] , [8] and [9]). That conjecture plays a central role in the problem of classifying finite thick geometries of type $C_n$ and $F₄$ ( see [4] , [7] and [8] ).In this paper we find very simple conditions that hold in a finite geometry of type $C₃$ with thick lines if and only if the geometry is either a building or flat. Then the previous conjecture can be restated so: prove that the conditions discussed in this paper are theorems in the case of finite geometries with thick lines.