Transverse spheres in flag manifolds

Recent results provide many flag manifolds whose transverse subsets cannot properly contain circles, which then restricts the corresponding class of Anosov subgroups. We complement those results by providing transverse 2-spheres (or larger) in all remaining cases of partial flag manifolds of split real Lie groups. We also find arbitrarily large maximally transverse spheres in certain full flag manifolds of split groups. This is joint work with Parker Evans.