# Metric Lie groups admitting dilations

@article{Donne2019MetricLG, title={Metric Lie groups admitting dilations}, author={Enrico Le Donne and Sebastiano Nicolussi Golo}, journal={arXiv: Metric Geometry}, year={2019} }

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 $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$.
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… Expand

#### 4 Citations

Polynomial and horizontally polynomial functions on Lie groups

- Mathematics
- 2020

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant… Expand

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

- Mathematics
- 2019

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in… Expand

Sublinear quasiconformality and the large-scale geometry of Heintze groups.

- Mathematics
- 2019

This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and… Expand

On rectifiable measures in Carnot groups: structure theory

- Mathematics
- 2020

In this paper we prove the one-dimensional Preiss' theorem in the first Heisenberg group $\mathbb H^1$. More precisely we show that a Radon measure $\phi$ on $\mathbb H^1$ with positive and finite… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

Isometries of nilpotent metric groups

- Mathematics
- 2016

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups.… Expand

Regularity properties of spheres in homogeneous groups

- Mathematics
- 2015

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with… Expand

On the rigidity of discrete isometry groups of negatively curved spaces

- Mathematics
- 1997

Abstract. We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees,… Expand

From homogeneous metric spaces to Lie groups

- Mathematics
- 2017

We study connected, locally compact metric spaces with transitive isometry groups. For all ε ∈ R, each such space is (1, ε)-quasi-isometric to a Lie group equipped with a left-invariant metric.… Expand

Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls

- Mathematics
- 2007

We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's… Expand

Carnot-Carathéodory spaces seen from within

- Physics
- 1996

Let V be a smooth manifold where we distinguish a subset H in the set of all piecewise smooth curves c in V. We assume that H is defined by a local condition on curves, i.e. if c is divided into… Expand

Hardy spaces on homogeneous groups

- Mathematics
- 1982

The object of this monograph is to give an exposition of the real-variable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to a… Expand

A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

- Mathematics
- 2018

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… Expand

A Tour of Subriemannian Geometries, Their Geodesics and Applications

- Mathematics
- 2006

Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of… Expand

Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

- Mathematics
- 2008

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a… Expand