Finite BN-pairs of rank 2, I

Abstract

One of the fundamental problems in Incidence Geometry is the classification of finite BN-pairs of rank $2$ (most notably those of type ${\mathsf{B}}_2$), without the use of the classification theorem for finite simple groups. In this paper, which is the first in a series, we classify finite BN-pairs of rank $2$ (and the buildings that arise) for which the associated parameters $\left(s,t\right)$ are powers of $2$, and such that the associated polygon has no proper thick ideal or full subpolygons. As a corollary, we obtain the complete classification of generalized octagons of order $\left(s,t\right)$ with $st$ a power of $2$, admitting a BN-pair. (For quadrangles and hexagons, this result will be obtained in part II.)