Rigidity of flag manifolds

Let N ⊂ GL(n,R) be the group of upper triangular matrices with 1s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms, and more generally Sobolev mappings with nondegenerate Pansu differential, are rigid when n ≥ 4; this settles the Regularity Conjecture for such groups. This result is deduced from a rigidity theorem for the manifold of complete flags in R. Similar results also hold in the complex and quaternion cases.

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