We have made significant use of Lie techniques in our non-smooth geometries. These techniques encompassed various aspects, such as utilizing the theory of free nilpotent groups and algebraic quotients to simplify metric properties. We employed differential calculus on Lie groups for metric calculations related to curve shortening, semisubgroups of Lie groups, and Lie algebra classifications. Metric geometry played a crucial role. We utilized submetries, Gromov-Hausdorff limits of geodesics and distances, and visual distances. Furthermore, control theory was instrumental in our investigations.
Our project made significant strides in addressing the problem of rectifiability of finite-perimeter sets and embeddings. While the debate continues regarding the strictest notion of intrinsic rectifiability, we established a covering-type result for sets with finite subRiemannian perimeter. This coverage was achieved through sets exhibiting a cone property, akin to classical Euclidean spaces. Notably, our work challenged the expectation of achieving stronger rectifiability through intrinsically differentiable hypersurfaces, owing to the construction of pathological examples by the PI and the research team. Importantly, the PI made substantial contributions to the theory of sets with constant normal (and, more broadly, monotone sets), finally determining whether certain non-Abelian Carnot groups can be biLipschitz embedded into the space of integrable real functions.
Regarding the growth of nilpotent groups and the shape of spheres in Carnot groups, the project made progress in all planned directions. The PI derived a sharper (though potentially not the sharpest) asymptotic expansion for the volume growth of Lie nilpotent groups equipped with general subFinsler metrics. The rectifiability of boundaries of metric balls in Carnot groups was established in groups of step 2, but there is growing skepticism regarding its validity in groups of step 3.
The PI provided a comprehensive answer to which homogeneous groups satisfy the Besicovitch Covering Property. However, the general case for arbitrary Lie groups remains unsolved. Our research on abnormal curves and the regularity of geodesics yielded several results, yet many questions remain unanswered. Currently, the strongest result says that tangents of subFinsler geodesics are geodesic when projected one step lower. While the outcome eliminates many pathologies via iteration, it doesn't provide new equations for subRiemannian geodesics nor does it establish their differentiability. Progress was made on the Sard Conjecture, particularly in groups of low step.
One of the most successful aspects of the project involved determining the algebraic properties of isometries between metric groups. The description of these properties was technical and resulted in a lengthy publication. However, it comprehensively elucidated the conditions under which two groups can be isometric, namely when they share the same real-shadow. We also achieved metric classifications of 3D groups and Heintze groups in low dimensions.
Another fruitful line of research focused on the conformal equivalence of visual metrics in strictly pseudoconvex domains. Inspired by Mostow's proof of his rigidity theorem, we offered an alternative proof of a result by Fefferman on smooth extensions of biholomorphic mappings between strictly pseudoconvex domains, showing that such extensions are conformal maps between subRiemannian manifolds.
Significant progress was also made in studying metrically homogeneous spaces. We achieved several characterizations of subFinsler geometries and metric Lie groups, including a quasi-Möbius characterization of invertible homogeneous metric spaces. A comprehensive description was provided for metric Lie groups that admit dilations, shedding light on locally compact homogeneous metric spaces that also admit dilations. Our work culminated in the classification known as the 'Cornucopia of Carnot Groups in Low Dimensions,' serving as a valuable resource for researchers exploring nilpotent and Carnot groups.
