On the Kirchheim-Magnani counterexample to metric differentiability

In Kirchheim-Magnani [7] the authors construct a left invariant distance ρ on the Heisenberg group such that the identity map id is 1-Lipschitz but it is not metrically differentiable anywhere. In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability. In fact we show that they construct something which fails shortly from being a dilatation structure. Dilatation structures have been introduced in [2]. These structures are related to conical group [3], which form a particular class of contractible groups and are a slight generalization of Carnot groups. Carnot groups, in particular the Heisenberg group, appear as infinitesimal models of sub-riemannian manifolds [1], [6]. In [5] we explain how the formalism of dilatation structures applies to sub-riemannian geometry. Further on we shall use the notations, definitions and results concerning dilatation structures, as found in [2], [3] or [5]. We shall construct a structure (H(1), ρ, δ̄) on H(1) which satisfies all axioms of a dilatation structure, excepting A3 and A4. We prove that for (H(1), ρ, δ̄) the axiom A4 implies A3. Finally we prove that A4 for (H(1), ρ, δ̄) is equivalent with id metrically differentiable from (H(1), d) to (H(1), ρ), where d is a left invariant CC distance. For other relations between dilatation structures and differentiability in metric spaces see [4].