Rank three geometry and positive curvature

TL;DR

This paper characterizes certain geometric structures called buildings of type C3 and shows that specific group actions on positively curved manifolds are equivalent to actions on rank one symmetric spaces, extending known classifications.

Contribution

It provides an axiomatic characterization of C3-type buildings and classifies cohomogeneity two polar actions of this type on positively curved manifolds.

Findings

Cohomogeneity two polar actions of type C3 are diffeomorphic to actions on rank one symmetric spaces.

Includes analysis of two actions on the Cayley plane with non-building C3 geometries.

Extends classification of group actions in positive curvature settings.

Abstract

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.