Hyperfield extensions, characteristic one and the Connes-Consani plane connection

2014/07/02 by Koen Thas, Thas, Koen
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR

paper · pdf · doi:10.48550/arxiv.1407.0607

25 pages; submitted (June 2014). arXiv admin note: text overlap with arXiv:1406.5447

arxiv created 2014/07/02 · arxiv updated 2014/07/03

Abstract

Inspired by a recent paper of Alain Connes and Catherina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces ("Singer actions"), we consider several fudamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields \mathbbF the plane P²(\mathbbF) = PG(2,\mathbbF) (and more generally the space Pⁿ(\mathbbF) = PG(n,\mathbbF)) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1, PG(n,\mathbbF_p) cannot admit Singer groups. One of the main results in characteristic 0, also as a corollary of a criterion which applies to many other fields, is that PG(m,ℝ) with m ≠ 0 a positive even integer, cannot admit Singer groups.

Citations

• The combinatorial-motivic nature of \mathbbF₁-schemes

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