  This text is dedicated to Jacques Tits's ideas on geometry over F1, the field
with one element. In a first part, we explain how thin Tits geometries surface
as rational point sets over the Krasner hyperfield, which links these ideas to
combinatorial flag varieties in the sense of Borovik, Gelfand and White and
F1-geometry in the sense of Connes and Consani. A completely novel feature is
our approach to algebraic groups over F1 in terms of an alteration of the very
concept of a group. In the second part, we study an incidence-geometrical
counterpart of (epimorphisms to) thin Tits geometries; we introduce and
classify all F1-structures on 3-dimensional projective spaces over finite
fields. This extends recent work of Thas and Thas on epimorphisms of projective
planes (and other rank 2 buildings) to thin planes.
