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Arkiv för Matematik


Lipschitz structure and minimal metrics on topological groups1
Pub. online: 5 September 2023      Type: Research Article     

1 The research was partially supported by a Simons Foundation Fellowship (Grant #229959) and by the NSF (DMS 1201295 & DMS 1464974). The author is grateful for very helpful conversations with I. Goldbring and for the detailed comments by the referees
Received
12 November 2016
Revised
2 July 2017
Published
5 September 2023

Abstract

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.

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Copyright
© 2018 by Institut Mittag-Leffler. All rights reserved

Keywords
metrisable groups left-invariant metrics Hilbert’s fifth problem Lipschitz structure

MSC
22A10 (primary) 03E15 (secondary)

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since February 2017
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