Towards the horizons of Tits’s vision : on band schemes, crowds and 𝔽1-structures

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.