Normed versus topological groups: Dichotomy and duality

14 N. H. Bingham and A. J. Ostaszewskigiving, by Theorem 2.8 and because ‖xy‖ ≥ ‖x‖ − ‖y‖,p ∗ (y) := lim sup x→∞p(xy)p(x) ≤ µ2 .The left-sided inequality is proved dually (interchanging the roles of the upper and lowerbounds on ‖π(x)‖ Y ).Remarks. 1. In the case of the general (µ-γ)-quasi-isometry, p exhibits the normedgroupsO-analogue of slow-variation; compare [BGT, Cor. 2.0.5 p. 65].2. When X = R the weaker boundedness property: “p ∗ (y) < ∞ on a large enough set ofys” implies that p satisfiesz d ≤ p ∗ (z) ≤ p ∗ (z) ≤ z c , (z ≥ Z)for some constants c, d, Z (so is extended regularly varying in the sense of [BGT, Ch. 2,2.2 p. 65]). Some generalizations are given in Theorems 7.10 and 7.11.3. We pause to consider briefly some classical examples. If X = H = R is construedadditively, so that e H = e X = 0 and ‖x‖ := |x − 0| = |x| in both cases, and withthe action tx denoting t + x, the function f(x) := |x| is not slowly varying, because(x + t) − x = t ↛ 0 = e H . On the other hand a multiplicative construction on H = R ∗ +,for which e H = 1 and ‖h‖ H := | log h|, but with X = R still additive and tx still meaningt + x, yields f as having slow variation (as in the Theorem 2.8), asf(tx)f(x) −1 = (x + t)/x → 1 = e H as x → ∞.We note that in this context the regularly varying functions h on X have h(tx)h(x) −1 =h(t + x) − h(x) → at, for some constant a.Note that, for X = H = R ∗ +, and with tx meaning t·x, since ‖x‖ = | log x| (as just noted)is the group-norm, we have heref(tx)f(x) −1 = ‖tx‖/‖x‖ =| log tx|| log x|=| log t + log x|| log x|→ 1 = e H , as x → ∞,which again illustrates the content of Theorem 2.7. Here the regularly varying functionsh(tx)h(x) −1 → e at , for some constant a. See [BGT, Ch. 1] for background on additiveand multiplicative formulations of regular variation in the classical setting of functionsf : G → H with G, H = R or R + .Definitions. 1. Say that ξ ∈ X is infinitely divisible if, for each positive integer n, thereis x with x n = ξ. (Compare Section 3.)2. Say that the infinitely divisible element ξ is embeddable if, for some one-parametersubgroup T in X, we have ξ ∈ T. When such a T exists it is unique (the elements ξ m/n ,for m, n integers, are dense in T ); we write T (ξ) for it.Clearly any element of a one-parameter subgroup is both infinitely divisible and embeddable.For results on this see Davies [D], Heyer [Hey], McCrudden [McC]. With these