On 3-nondegenerate CR manifolds in dimension 7 (I): the transitive case

Abstract

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka's conjecture on the symmetry dimension bound for hypersurfaces in ${\mathbb{C}}^4$. We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.

40 pages, v3: Final version to appear on J. Reine Angew. Math. The supplementary files are not reproduced in this version, but they are equally relevant and can be accessed through v1