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

TL;DR

This paper explores the geometric and algebraic structures over the field with one element (F1), linking Tits geometries to hyperfields and classifying F1-structures on projective spaces, advancing F1-geometry theory.

Contribution

It introduces a novel approach to algebraic groups over F1 and classifies F1-structures on 3D projective spaces, extending previous work on thin geometries.

Findings

Thin Tits geometries as rational point sets over Krasner hyperfield

Classification of F1-structures on 3D projective spaces

Extension of epimorphism results to thin planes

Abstract

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.