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

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 $\mathbb{F}$ the plane $\mathbf{P}^2(\mathbb{F}) = \mathbf{PG}(2,\mathbb{F})$ (and more generally the space $\mathbf{P}^n(\mathbb{F}) = \mathbf{PG}(n,\mathbb{F})$) admits a Singer group, and show, e.g., that for any prime $p$ and any positive integer $n > 1$, $\mathbf{PG}(n,\overline{\mathbb{F}_p})$ 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 $\mathbf{PG}(m,\mathbb{R})$ with $m \ne 0$ a positive even integer, cannot admit Singer groups.Comment: 25 pages; submitted (June 2014). arXiv admin note: text overlap with arXiv:1406.544

Quotients of fake projective planes

Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to \bbZ/3\bbZ, \bbZ/7\bbZ, $7:3$, or (\bbZ/3\bbZ)^2, where $7:3$ is the unique non-abelian group of order 21. Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X/G$ and its minimal resolution.Comment: 16 pages, with minor change of the expositio