Contractible groups and linear dilatation structures

TL;DR

This paper explores the concept of dilatation structures on metric spaces, characterizes contractible groups through these structures, and distinguishes two types of linearity within this framework.

Contribution

It introduces a characterization of contractible groups using dilatation structures and clarifies the notions of linearity for functions and structures within this context.

Findings

Contractible groups are characterized via dilatation structures.

Linear dilatation structures correspond to normed contractible groups.

The paper distinguishes two types of linearity: of functions and of structures.

Abstract

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.