Metric Lie groups admitting dilations

Enrico Le Donne Department of Mathematics and Statistics, University of Jyväskylä, Finland; and Dipartimento di Matematica, Università di Pisa, Italy Sebastiano Nicolussi Golo Dipartimento di Matematica, Università di Padova, Italy

Differential Geometry Algebraic Topology and General Topology mathscidoc:2203.10001

Arkiv for Matematik, 59, (1), 125-163, 2021.5
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,∞)→Aut(G), λ↦δλ, so that d(δ_λx,δ_λy)=λd(x,y), for all x,y∈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 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,∞). 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.
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@inproceedings{enrico2021metric,
  title={Metric Lie groups admitting dilations},
  author={Enrico Le Donne, and Sebastiano Nicolussi Golo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311104159671880955},
  booktitle={Arkiv for Matematik},
  volume={59},
  number={1},
  pages={125-163},
  year={2021},
}
Enrico Le Donne, and Sebastiano Nicolussi Golo. Metric Lie groups admitting dilations. 2021. Vol. 59. In Arkiv for Matematik. pp.125-163. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311104159671880955.
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