From homogeneous metric spaces to Lie groups
From homogeneous metric spaces to Lie groups
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从齐次度量空间到李群
DOI：
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发表时间：
2017
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通讯作者：
A. Ottazzi
中科院分区：
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作者：
M. Cowling;Ville Kivioja;E. Donne;S. Golo;A. Ottazzi
本文研究了连通的局部紧度量空间和可迁等距群。对于所有ε ∈ R，每个这样的空间对于一个具有左不变度量的李群是（1，ε）-拟等距的。进一步地，每个度量李群与可解李群是（1，C）-拟等距的，并且每个单连通度量李群与可解紧度量李群是（1，C）-拟等距同胚的。虽然任何可收缩李群都可以等距于一个可解群，但只有那些可解的并且是（R）型的李群才可以等距于一个幂零李群，在这种情况下，幂零群是这个群的零影。最后，我们给出了度量李群存在自守伸缩的一个完备度量刻画。这些与度量空间是局部紧的，连通的，齐次的，并允许度量膨胀。
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. Further, every metric Lie group is (1, C)-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is (1, C)-quasi-isometrically homeomorphic to a solvable-bycompact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation.