Normed versus topological groups: Dichotomy and duality

Normed groups 412. Further recall that a group-norm is N-homogeneous if it is n-homogeneous for eachn ∈ N, i.e. for each n ∈ N, ‖x n ‖ = n‖x‖for each x. Thus if ξ n = x, then ‖ξ‖ = 1 n‖x‖ and,as ξ m = x m/n , we have m n ‖x‖ = ‖xm/n ‖, i.e. for rational q > 0 we have q‖x‖ = ‖x q ‖.Theorem 3.20 below relates the Lischitz property of a norm to local behaviour. Oneshould expect local behaviour to be critical, as asymptotic properties are trivial, since bythe triangle inequalitylim ‖x‖→∞‖x‖ g‖x‖ = 1.As this asserts that ‖x‖ g is slowly varying (see Section 2) and ‖x‖ g is continuous, theUniform Convergence Theorem (UCT) applies (see [BOst-TRI]; for the case G = R see[BGT]), and so this limit is uniform on compact subsets of G. Theorem 3.21 identifiescircumstances when a group-norm on G has the Lipschitz property and Theorem 3.22considers the Lipschitz property of the supremum norm in H u (X).On a number of occasions, the study of group-norm behaviour is aided by the presenceof the following property. Its definition is motivated by the notion of an ‘invariantconnected metric’ as defined in [Var, Ch. III.4] (see also [NSW]). The property expressesscale-comparability between word-length and distance, in keeping with the key notion ofquasi-isometry.Definition (Word-net). Say that a normed group G has a group-norm ‖.‖ with avanishingly small word-net (which may be also compactly generated, as appropriate) if,for any ε > 0, there is η > 0 such that, for all δ with 0 < δ < η there is a set (a compactset) of generators Z δ in B δ (e) and a constant M δ such that, for all x with ‖x‖ > M δ ,there is some word w(x) = z 1 ...z n(x) using generators in Z δ with ‖z i ‖ = δ(1 + ε i ), with|ε i | < ε, whereandSay that the word-net is global if M δ = 0.d(x, w(x)) < δ1 − ε ≤ n(x)δ‖x‖ ≤ 1 + ε.Remarks. 1. R d has a vanishingly small compactly generated global word-net and henceso does the sequence space l 2 .2. An infinitely divisible group X with an N-homogenous norm has a vanishingly smallglobal word-net. Indeed, given δ > 0 and x ∈ X take n(x) = ‖x‖/δ, then if ξ n = x wehave ‖x‖ = n‖ξ‖, and so ‖ξ‖ = δ and n(x)δ/‖x‖ = 1.