Arkiv för Matematik

Volume 56 (2018)

Number 1

Lipschitz structure and minimal metrics on topological groups

Pages: 185 – 206

DOI: https://dx.doi.org/10.4310/ARKIV.2018.v56.n1.a11

Author

Christian Rosendal (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.; and Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

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.

Keywords

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

2010 Mathematics Subject Classification

Primary 22A10. Secondary 03E15.

Received 12 November 2016

Received revised 2 July 2017

Accepted 8 September 2017

Published 30 April 2018