Lipschitz structure and minimal metrics on topological groups

Christian Rosendal University of Illinois & University of Maryland

Group Theory and Lie Theory mathscidoc:1912.43011

Arkiv for Matematik, 56, (1), 185-206, 2018
We discuss the problem of deciding when a metrisable topological group G has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on G, that we characterise intrinsically in terms of a linear growth condition on powers of group elements. Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry. In turn, minimal metrics connect with Hilbert’s fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a 1-parameter subgroup.
metrisable groups, left-invariant metrics, Hilbert’s fifth problem, Lipschitz structure
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@inproceedings{christian2018lipschitz,
  title={Lipschitz structure and minimal metrics on topological groups},
  author={Christian Rosendal},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204100102977494567},
  booktitle={Arkiv for Matematik},
  volume={56},
  number={1},
  pages={185-206},
  year={2018},
}
Christian Rosendal. Lipschitz structure and minimal metrics on topological groups. 2018. Vol. 56. In Arkiv for Matematik. pp.185-206. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204100102977494567.
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