Rank three geometry and positive curvature

An axiomatic characterization of buildings of type \CC_3 due to Tits is used to prove that any cohomogeneity two polar action of type \CC_3 on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated \CC_3 type geometry is not covered by a building

Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.Comment: 35 page

Partition dimension of projective planes

We determine the partition dimension of the incidence graph G(Πq) of the projective plane Πq up to a constant factor 2 as (2+o(1))log2q≤pd(G(Πq))≤(4+o(1))log2q. © 2017 Elsevier Lt

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