Normed versus topological groups: Dichotomy and duality

1. IntroductionGroup-norms, which behave like the usual vector norms except that scaling is restrictedto the basic scalars of group theory (the units ±1 in an abelian context and the exponents±1 in the non-commutative context), have played a part in the early development of topologicalgroup theory. They appear naturally in the study of groups of homeomorphisms.Although ubiquitous, they lack a clear and unified exposition. This lack is our motivationhere, since they offer the right context for the recent theory of topological regular variation.This extends the classical theory (for which see, e.g. [BGT]) from the real line tometrizable topological groups. Normed groups are just groups carrying a right-invariantmetric. The basic metrization theorem for groups, the Birkhoff-Kakutani Theorem of1936 ([Bir], [Kak], see [Kel, Ch.6 Problems N-R], [Klee], [Bour, Part 2, Section 4.1], and[ArMa], compare also [Eng, Exercise 8.1.G and Th. 8.1.21]), is usually stated as assertingthat a first-countable Hausdorff group has a right-invariant metric. It is properly speakinga ‘normability’ theorem in the style of Kolmogorov’s Theorem ([Kol], or [Ru, Th. 1.39]; inthis connection see also [Jam], where strong forms of connectedness are used in an abeliansetting to generate norms), as we shall see below. Indeed the metric construction in [Kak]is reminiscent of the more familiar construction of a Minkowski functional (for whichsee [Ru, Sect. 1.33]), but is implicitly a supremum norm – as defined below; in Rudin’sderivation of the metric (for a topological vector space setting, [Ru, Th. 1.24]) this normis explicit. Early use by A. D. Michal and his collaborators was in providing a canonicalsetting for differential calculus (see the review [Michal2] and as instance [JMW]) andincluded the noteworthy generalization of the implicit function theorem by Bartle [Bart](see Th. 10.10). In name the group-norm makes an explicit appearance in 1950 in [Pet1]in the course of his classic closed-graph theorem (in connection with Banach’s closedgraphtheorem and the Banach-Kuratowski category dichotomy for groups). It reappearsin the group context in 1963 under the name ‘length function’, motivated by word length,in the work of R. C. Lyndon [Lyn2] (cf. [LynSch]) on Nielsen’s Subgroup Theorem, thata subgroup of a free group is a free group. (Earlier related usage for function spaces isin [EH].) The latter name is conventional in geometric group theory despite the parallelusage in algebra (cf. [Far]) and the recent work on norm extension (from a normalsubgroup) of Bökamp [Bo].When a group is topologically complete and also abelian, then it admits a metric whichis bi-invariant, i.e. is both right- and left-invariant, as [Klee] first showed (in course ofsolving a problem of Banach). In Section 3 we characterize non-commutative groups thathave a bi-invariant metric, a context of significance for the calculus of regular variation[1]