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We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
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Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnors sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups.
We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in Euclidean spaces, as introduced by Chen. The constructive limit space has the universal property in the category of pointed metric spaces with 1-Lipschitz maps. In the general setting some metric properties are discussed such as the existence of geodesics and lifts. The notion of submetry will play a crucial role. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, such limit space is in correspondence with the space of signatures of rectifiable paths in $mathbb R^n$. Hambly-Lyonss result on the uniqueness of signature implies that this space is a geodesic metric tree that brunches at every point with infinite valence. As a particular consequence we deduce that every path in $mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Caratheodory spaces and of nilpotent metric Lie groups.
This paper contributes to the generalization of Rademachers differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce the notion of metric scalable groups which are our infinite-dimensional analogues of Carnot groups. The groups in which we will mostly be interested are the ones that admit a dense increasing sequence of (finite-dimensional) Carnot subgroups. In fact, in each of these spaces we show that every Lipschitz function has a point of G^{a}teaux differentiability. We provide examples and criteria for when such Carnot subgroups exist. The proof of the main theorem follows the work of Aronszajn and Pansu.
In this paper, we construct Holder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $mathbb{H}$. Pansu and Gromov observed that any surface embedded in $mathbb{H}$ has Hausdorff dimension at least 3, so there is no $alpha$-Holder embedding of a surface into $mathbb{H}$ when $alpha>frac{2}{3}$. Zust improved this result to show that when $alpha>frac{2}{3}$, any $alpha$-Holder map from a simply-connected Riemannian manifold to $mathbb{H}$ factors through a metric tree. In the present paper, we show that Zusts result is sharp by constructing $(frac{2}{3}-epsilon)$-Holder maps from $D^2$ and $D^3$ to $mathbb{H}$ that do not factor through a tree. We use these to show that if $0<alpha < frac{2}{3}$, then the set of $alpha$-Holder maps from a compact metric space to $mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $mathbb{R}^3$ to $mathbb{H}$ with Holder exponents arbitrarily close to $frac{2}{3}$.
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