Abstract

This text is dedicated to Jacques Tits’s ideas on geometry over ${\mathbb{𝔽}}_1$, 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 ${\mathbb{𝔽}}_1$-geometry in the sense of Connes and Consani. A novel feature is our approach to algebraic groups over ${\mathbb{𝔽}}_1$ 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 ${\mathbb{𝔽}}_1$-structures on $3$-dimensional projective spaces over finite fields. This extends recent work of J. A. and K. Thas on epimorphisms of projective planes (and other rank $2$ buildings) to thin planes.

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