Lipschitz structure and minimal metrics on topological groups
Volume 56, Issue 1 (2018), pp. 185–206
Pub. online: 7 October 2022
Type: Article
Open Access
Open Access
Received
12 November 2016
12 November 2016
Revised
2 July 2017
2 July 2017
Published
7 October 2022
7 October 2022
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.