[Paper Review] Metric Lie groups admitting dilations

Enrico Le Donne, Sebastiano Nicolussi Golo|arXiv (Cornell University)|Jan 8, 2019

Geometric Analysis and Curvature Flows20 references8 citations

TL;DR

This paper investigates left-invariant distances on Lie groups that admit a one-parameter group of metric dilations, proving that such distances are always admissible (induce the manifold topology) and characterizing the algebraic conditions under which they exist. The key contribution is a complete characterization of derivations that generate such dilations, showing that existence implies the Lie group is connected, simply connected, and nilpotent, with spectral constraints on the infinitesimal generator.

ABSTRACT

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty) ightarrow\mathtt{Aut}(G)$, $λ\mapstoδ_λ$, so that $ d(δ_λx,δ_λy) = λd(x,y)$, for all $x,y\in G$ and all $λ>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,\infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.

Motivation & Objective

• To characterize left-invariant distances on Lie groups that admit a one-parameter group of metric dilations.
• To determine the necessary and sufficient algebraic conditions on the infinitesimal generator of such dilations for the existence of an associated distance.
• To show that any admissible left-invariant distance with at least one nontrivial dilating automorphism is biLipschitz equivalent to one with a one-parameter group of dilations.
• To identify which automorphisms of a Lie group can arise as dilating automorphisms for some admissible distance.
• To characterize metric Lie groups with one-parameter groups of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors.

Proposed method

• Use the theory of one-parameter groups of Lie automorphisms to define $A$-homogeneous distances via a derivation $A \in \mathrm{Der}(\mathfrak{g})$.
• Prove that $A$-homogeneous distances are admissible (induce the manifold topology) by analyzing the action of $\delta_\lambda = \lambda^A$ on the group.
• Characterize the existence of $A$-homogeneous distances using the grading induced by $A$'s generalized eigenspaces and spectral conditions.
• Apply the Jordan-Chevalley decomposition to separate the semisimple and nilpotent parts of $A$, and analyze the resulting structure.
• Use biLipschitz equivalence to reduce general $A$-homogeneous distances to those with real spectrum, particularly when $A$ is diagonalizable on $\mathbb{C}$.
• Construct counterexamples to show that not all $A$-homogeneous distances arise from real-spectrum generators, using non-diagonalizable $A$ with complex eigenvalues.

Experimental results

Research questions

• RQ1 Under what algebraic conditions on a derivation $A$ does there exist an $A$-homogeneous distance on a Lie group $G$?
• RQ2 Can every admissible left-invariant distance on a Lie group with a nontrivial dilating automorphism be biLipschitz equivalent to one admitting a one-parameter group of dilations?
• RQ3 Which automorphisms of a Lie group can be dilating automorphisms for some admissible distance?
• RQ4 What is the intrinsic geometric characterization of metric Lie groups admitting a one-parameter group of dilating automorphisms?
• RQ5 Are there metric spaces with metric dilations that are not isometric to any $A$-homogeneous space with real-spectrum generator $A$?

Key findings

• Any $A$-homogeneous distance on a Lie group $G$ is admissible, meaning it induces the manifold topology, and thus $G$ must be connected.
• An $A$-homogeneous distance exists on $G$ if and only if $G$ is connected and simply connected, $V_t = 0$ for all $t < 1$, and $A|_{V_1}$ is diagonalizable over $\mathbb{C}$.
• If the spectrum of $A$ lies on the line $1 + i\mathbb{R}$, then any $A$-homogeneous distance is also $\mathrm{Id}$-homogeneous, i.e., it is a norm on an abelian group.
• There exist locally compact, isometrically homogeneous metric spaces with metric dilations that are not isometric to any $A$-homogeneous space with real-spectrum generator $A$, demonstrating the necessity of complex spectra.
• Any admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to a distance admitting a one-parameter group of dilating automorphisms with generator having spectrum in $[1, \infty)$.
• Metric Lie groups admitting a one-parameter group of dilating automorphisms are precisely the only locally compact, isometrically homogeneous metric spaces with metric dilations for all scaling factors, and they arise as tangents of doubling metric spaces with unique tangents.