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# Arkiv för Matematik

## Volume 59 (2021)

### Number 1

### Metric Lie groups admitting dilations

Pages: 125 – 163

DOI: https://dx.doi.org/10.4310/ARKIV.2021.v59.n1.a5

#### Authors

#### 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 $(0,\infty) \to \operatorname{Aut}(G), \lambda \mapsto \delta_\lambda$, so that $d (\delta_\lambda x, \delta_\lambda y) = \lambda d(x, y),$ for all $x,y \in G$ and all $\lambda \gt 0$.

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 bi-Lipschitz 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.

#### Keywords

homothety, metric Lie group, grading

#### 2010 Mathematics Subject Classification

53C30, 54E40, 54E45

E.L.D. has been partially supported by the Academy of Finland (grant 288501 ‘*Geometry of subRiemannian groups*’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘*Geometry of Metric Groups*’). S.N.G. has been partially supported by the European Unions Seventh Framework Programme, Marie Curie Actions-Initial Training Network, under grant agreement n. 607643, “Metric Analysis For Emergent Technologies (MAnET)”, by the EPSRC Grant “Sub-Elliptic Harmonic Analysis” (EP/P002447/1), and by University of Padova STARS Project “Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New”.

Received 30 January 2020

Received revised 28 July 2020

Accepted 7 August 2020

Published 4 May 2021