Some results on Tits' geometries of type $F₄$

Abstract

It is known that all finite thick geometries of type $C_n (n≥ 4)$ with known parameters are buildings (see [10] and [6]).Several facts suggest the conjecture that the same holds in general.Moreover, a finite thick geometry of type $F₄$ with known parameters is a building unless its parameters are as below (see[6]). It is sensible to conjecture that all finite thick geometries of type $F₄$ are buildings.I am not able to prove this conjecture.But I collect in this paper some partial results related to this problem.They improve other results given in [9] and [6].