Contraction groups and linear dilatation structures

A dilatation structure on a metric space, introduced in [3], is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. A metric space (X, d) which admits a strong dilatation structure (definition 3.2) has a metric tangent space at any point x ∈ X (theorem 4.2), and any such metric tangent space has an algebraic structure of a conical group (theorem 4.3). Conical groups are particular examples of contraction groups. The structure of contraction groups is known in some detail, due to Siebert [8], Wang [9], Glockner and Willis [5], Glockner [4] and references therein. By a classical result of Siebert [8] proposition 5.4, we can characterize the algebraic structure of the metric tangent spaces associated to dilatation structures of a certain kind: they are Carnot groups, that is simply connected Lie groups whose Lie algebra admits a positive graduation (corollary 4.7). Carnot groups appear in many situations, in particular in relation with subriemannian geometry cf. Belläıche [1], groups with polynomial growth cf. Gromov [6], or Margulis type rigidity results cf. Pansu [7]. It is part of the author program of research to show that dilatation structures are natural objects in all these mathematical subjects. In this respect the corollary 4.7 represents a generalization of difficult results in sub-riemannian geometry concerning the structure of the metric tangent space at a point of a regular subriemannian manifold. Linearity is also a property which can be explained with the help of a dilatation structure. In the second section of the paper we explain why linearity can be casted in terms of dilatations. In this paper we show that we can speak about two kinds