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Towards the horizons of Tits's vision: on band schemes, crowds and $\mathbb{F}_1$-structures

Oliver Lorscheid and Koen Thas

Vol. 20 (2023), No. 2-3, 353–394
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 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 J. A. and K. Thas on epimorphisms of projective planes (and other rank 2 buildings) to thin planes.

Keywords
Jacques Tits, field with one element, F1-geometry, generalized polygons
Mathematical Subject Classification
Primary: 14Kxx, 14Lxx, 51E20, 51E24, 51Exx
Milestones
Received: 17 February 2023
Revised: 14 August 2023
Accepted: 19 August 2023
Published: 13 September 2023
Authors
Oliver Lorscheid
Bernoulli Institute
University of Groningen
9747 AG Groningen
Netherlands
and
IMPA
CEP 22460-320
Rio de Janeiro
Brazil
Koen Thas
Department of Mathematics: Algebra and Geometry
Ghent University
9000 Ghent
Belgium