Normed versus topological groups: Dichotomy and duality

Normed groups 13More generally, for T a one-parameter subgroup of X, any sub-additive Baire functionp : X → R ∗ + with‖p‖ T := lim x∈T, ‖x‖→∞p(x)‖x‖ > 0is multiplicatively slowly varying. (The limit exists by the First Limit Theorem for Bairesubadditive functions, see [BOst-GenSub].)Proof. By Corollary 2.6, for x ≠ e,1 − ‖t‖‖x‖ ≤ ‖tx‖‖x‖ ≤ 1 + ‖t‖‖x‖ ,which implies slow variation. We regard p as mapping to R ∗ +, the strictly positive reals(since p(x) = 0 iff x = e X ). Taking h = p and µ = ‖p‖ T > 0, the assertion follows fromthe Comparison Criterion (Th. 2.7) above (with g(x) = ‖x‖). Explicitly, for x ≠ e,p(xy)p(x) = p(xy)‖xy‖ · ‖xy‖‖x‖ · ‖x‖p(x) → ‖p‖ 1T · 1 · = 1.‖p‖ TCorollary 2.9. If π : X → Y is a group homomorphism and ‖ · ‖ Y is (1-γ)-quasiisometricto ‖ · ‖ X under the mapping π, then the subadditive function p(x) = ‖π(x)‖ Yis slowly varying. For general (µ-γ)-quasi-isometry the function p satisfieswhereµ −2 ≤ p ∗ (z) ≤ p ∗ (z) ≤ µ 2 ,p ∗ (z) = lim sup ‖x‖→∞ p(zx)p(x) −1 p ∗ (z) = lim inf ‖x‖→∞ p(zx)p(x) −1 .Proof. Subadditivity of p follows from π being a homomorphism, since p(xy) = ‖π(xy)‖ Y= ‖π(x)π(y)‖ Y ≤ ‖π(x)‖ Y + ‖π(y)‖ Y . Assuming that, for µ = 1 and γ > 0, the norm‖ · ‖ Y is (µ-γ)-quasi-isometric to ‖ · ‖ X , we have, for x ≠ e,So1 − γ ≤ p(x) ≤ 1 −γ .‖x‖ X ‖x‖ X ‖x‖ Xlim ‖x‖→∞p(x)‖x‖ = 1 ≠ 0,and the result follows from the Comparison Criterion (Th. 2.7) and Theorem 2.5.If, for general µ ≥ 1 and γ > 0, the norm ‖ · ‖ Y is (µ-γ)-quasi-isometric to ‖ · ‖ X , wehave, for x ≠ e,µ −1 − γ ≤ p(x) ≤ µ −γ .‖x‖ X ‖x‖ X ‖x‖ XSo for y fixedp(xy)p(x) = p(xy)‖xy‖ · ‖xy‖‖x‖ · ‖x‖p(x) ≤ (µ − γ‖xy‖ X)· ‖xy‖‖x‖ ·(µ −1 − γ‖x‖ X) −1,