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Metric Lie groups admitting dilations
Arkiv för Matematik ( IF 0.7 ) Pub Date : 2021-04-01 , DOI: 10.4310/arkiv.2021.v59.n1.a5
Enrico Le Donne 1 , Sebastiano Nicolussi Golo 2
Affiliation  

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.

中文翻译:

允许扩张的度量李群

我们考虑一个李群$ G $的左不变距离$ d $,其性质是存在一个与李算子同构$(0,\ infty)\到\ operatorname {Aut}(G),\ lambda \ mapsto \ delta_ \ lambda $,因此$ d(\ delta_ \ lambda x,\ delta_ \ lambda y)= \ lambda d(x,y),$表示所有$ x,y \ in G $和所有$ \ lambda \ gt 0 $。首先,我们证明所有这样的距离都是可以接受的,也就是说,它们会诱发流形拓扑。其次,我们描述了Lie自同构的乘法一参数组,这些组是根据它们的无穷小生成器的代数性质对某些左不变距离进行扩张的。第三,我们表明,具有至少一个非平凡扩张自同构的Lie群上的容许左不变距离是bi-Lipschitz等效值,它等同于允许一个参数组的扩张自同构性。此外,可以选择无穷小生成器以使频谱具有$ [1,\ infty)$。第四,我们刻画了李群的自同构性,它们是在一定容许距离上的扩张自同构性。最后,我们将接纳自扩张同构的一个参数组的度量李群定性为唯一具有所有因子度量扩张的局部紧致,等距均匀的度量空间。这样的度量空间显示为具有唯一切线的加倍度量空间的切线。具有所有因子的度量膨胀的等距齐次度量空间。这样的度量空间显示为具有唯一切线的加倍度量空间的切线。具有所有因子的度量膨胀的等距齐次度量空间。这样的度量空间显示为具有唯一切线的加倍度量空间的切线。
更新日期:2021-05-05
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