The Connes–Consani plane connection

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 FF the plane P2(F)=PG(2,F)P2(F)=PG(2,F) (and more generally the space Pn(F)=PG(n,F)Pn(F)=PG(n,F)) admits a Singer group, and show, e.g., that for any prime p   and any positive integer n>1n>1, PG(n,Fp‾) cannot admit Singer groups (Fp‾ an algebraic closure of FpFp). 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)PG(m,R) with m≠0m≠0 a positive even integer, cannot admit Singer groups.