Linear dilatation structures and inverse semigroups

A dilatation structure encodes the approximate self-similarity of a metric space. A metric space (X, d) which admits a strong dilatation structure (definition 2.2) has a metric tangent space at any point x ∈ X (theorem 4.1), and any such metric tangent space has an algebraic structure of a conical group (theorem 4.2). Particular examples of conical groups are Carnot groups: these are simply connected Lie groups whose Lie algebra admits a positive graduation. The dilatation structures associated to conical (or Carnot) groups are linear, in the sense of definition 5.3. Thus conical groups are the right generalization of normed vector spaces, from the point of view of dilatation structures. We prove that for dilatation structures linearity is equivalent to a statement about the inverse semigroup generated by the family of dilatations forming a dilatation structure on a metric space. The result is new for Carnot groups and the proof seems to be new even for the particular case of normed vector spaces.