A construction of one-dimensional affine flag-transitive linear spaces

AbstractThe finite flag-transitive linear spaces which have an insoluble automorphism group were given a precise description in [Francis Buekenhout, Anne Delandtsheer, Jean Doyen, Peter B. Kleidman, Martin W. Liebeck, Jan Saxl, Linear spaces with flag-transitive automorphism groups, Geom. Dedicata 36 (1) (1990) 89–94], and their classification has recently been completed (see [Martin W. Liebeck, The classification of finite linear spaces with flag-transitive automorphism groups of affine type, J. Combin. Theory Ser. A 84 (2) (1998) 196–235] and [Jan Saxl, On finite linear spaces with almost simple flag-transitive automorphism groups, J. Combin. Theory Ser. A 100 (2) (2002) 322–348]). However, the remaining case where the automorphism group is a subgroup of one-dimensional affine transformations has not been classified and bears a variety of known examples. Here we give a construction of new one-dimensional affine flag-transitive linear spaces via the André/Bruck–Bose construction applied to transitive line-spreads of projective space

Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie

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

Pairwise transitive 2-designs

We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples arise from affine geometry and the point-line geometry of projective spaces, as well as several sporadic examples.Comment: 28 pages, updated after review proces

Countable locally 2-arc-transitive bipartite graphs

We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These examples are obtained by gluing together copies of incidence graphs of semilinear spaces, satisfying a certain symmetry property, in a tree-like way. In one case we show how the classification problem for that family relates to the problem of determining a certain family of highly arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page

Doubly transitive lines II: Almost simple symmetries

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper, the second in a series, classifies those lines that exhibit almost simple symmetries. To perform this classification, we introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group