The Connes-Consani plane connection

摘要

Inspired by a recent paper of Alain Connes and Caterina 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 fundamental 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 F the plane P-2(F) = PG(2,F) (and more generally the space P-n (F) = PG(n, F)) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1, PG(n, (F) over bar (p)) cannot admit Singer groups ((F) over bar (p) an algebraic closure of F-p). One of the main results in characteristic 0, which is a corollary of a criterion which applies to many other fields, is that PG(m, R) with m not equal 0 a positive even integer, cannot admit Singer groups. (C) 2016 Elsevier Inc. All rights reserved.