Transverse Spheres in Flag Manifolds

TL;DR

This paper classifies partial flag manifolds with maximally transverse circles and spheres, revealing new geometric structures and subgroup properties in split real Lie groups using topological and algebraic methods.

Contribution

It completes the classification of flag manifolds with maximally transverse circles and constructs higher-dimensional transverse spheres using spinors and $K$-theory.

Findings

Classification of all partial flag manifolds with maximally transverse circles.

Existence of arbitrarily large transverse spheres in certain full flag manifolds.

Identification of fibrations by maximally transverse 3-spheres in specific types.

Abstract

For some partial flag manifolds of semisimple real Lie groups, including many full flag manifolds, transverse circles are known to be locally maximally transverse. We complete the classification of all partial flag manifolds of split real Lie groups with this property. As a consequence, $\{7\}$-Anosov subgroups of split $E_7$ are virtually free or surface groups. On the other hand, using spinors, we find transverse spheres of arbitrarily large dimension in certain full flag manifolds of Cartan-Killing types $A,B,D$. These transverse spheres are verified to be maximally transverse with tools from topological $K$-theory. The aforementioned classification follows from constructions of transverse $m$-spheres, $m \geq 2$, that complement the previously known restrictions as well as the new $E_7$ restriction. Additionally, when $G$ is split of type $G_2, B_3,$ or $D_4$, the full flag manifold admits a fibration by maximally transverse 3-spheres.