One of the fundamental problems in Incidence Geometry is the classification of finite BN-pairs of
rank
(most notably
those of type
),
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
(and the buildings that arise) for which the associated parameters
are powers
of
,
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
with
a power
of
,
admitting a BN-pair. (For quadrangles and hexagons, this result will be obtained in
part II.)
