Normed versus topological groups: Dichotomy and duality

Normed groups 11Corollary 2.4. For π a homomorphism, the normed groups X, Y are (µ-γ)-quasiisometricunder π for the corresponding metrics iff the associated norms are (µ-γ)-quasiequivalent,i.e.1µ ‖x‖ X − γ ≤ ‖π(x)‖ Y ≤ µ‖x‖ X + γ (a, b ∈ X),d Y (y, π[X]) ≤ γ (y ∈ Y ).Proof. This follows from π(e X ) = e Y and π(xy −1 ) = π(x)π(y) −1 .Remark. Note that p(x) = ‖π(x)‖ Y is subadditive and bounded at x = e. It will followthat p is locally bounded at every point when we later prove Lemma 4.3.The following result (which we use in [BOst-TRII]) clarifies the relationship betweenthe conjugate metrics and the group structure. We define the ε-swelling of a set K in ametric space X for a given (e.g. right-invariant) metric d X , to beB ε (K) := {z : d X (z, k) < ε for some k ∈ K} = ⋃ k∈K B ε(k)and for the conjugate (resp. left-invariant) case we can write similarlyWe write B ε (x 0 ) for B ε ({x 0 }), so that˜B ε (K) := {z : ˜d X (z, k) < ε for some k ∈ K}.B ε (x 0 ) := {z : ‖zx −10 ‖ < ε} = {wx 0 : w = zx −10 , ‖w‖ < ε} = B ε(e)x 0 .When x 0 = e X , the ball B ε (e X ) is the same under either of the conjugate metrics, asB ε (e X ) := {z : ‖z‖ < ε}.Proposition 2.5. (i) In a locally compact group X, for K compact and for ε > 0 smallenough so that the closed ε-ball B ε (e X ) is compact, the swelling B ε/2 (K) is pre-compact.(ii) B ε (K) = {wk : k ∈ K, ‖w‖ X < ε} = B ε (e X )K, where the notation refers toswellings for d X a right-invariant metric; similarly, for ˜d X , the conjugate metric, ˜B ε (K)= KB ε (e X ).Proof. (i) If x n ∈ B ε/2 (K), then we may choose k n ∈ K with d(k n , x n ) < ε/2. Withoutloss of generality k n converges to k. Thus there exists N such that, for n > N,d(k n , k) d X (z, k) =d X (zk −1 , e), then, putting w = zk −1 , we have z = wk ∈ B ε (K).