Contractible groups and linear dilatation structures

A dilatation structure on a metric space, arXiv:math/0608536v4, 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. Linearity is also a property which can be explained with the help of a dilatation structure. In this paper we show that we can speak about two kinds of linearity: the linearity of a function between two dilatation structures and the linearity of the dilatation structure itself. Our main result here is a characterization of contractible groups in terms of dilatation structures. To a normed conical group (normed contractible group) we can naturally associate a linear dilatation structure. Conversely, any linear and strong dilatation structure comes from the dilatation structure of a normed contractible group.

💡 Research Summary

The paper investigates the interplay between dilatation structures—a framework that sits between group theory and differential geometry—and contractible (or conical) groups equipped with a norm. A dilatation structure on a metric space ((X,d)) consists of a family of contraction maps (\delta^{x}{\varepsilon}:X\to X) indexed by a base point (x) and a scale parameter (\varepsilon>0). These maps satisfy a set of axioms (A0–A4) that encode how different dilatations interact, how they approximate the identity as (\varepsilon\to0), and how they behave with respect to the metric. In particular, axiom A2 expresses a “scale‑composition” rule (\delta^{x}{\varepsilon}\circ\delta^{\delta^{x}{\varepsilon}y}{\mu}=\delta^{x}_{\varepsilon\mu}), which captures an intrinsic self‑similarity of the space.

The author distinguishes two notions of linearity. The first is linearity of a map between two dilatation structures: a function (f:X\to Y) is linear if for every point (x\in X) and every scale (\varepsilon) we have
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