# Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

@article{Buliga2008InfinitesimalAG, title={Infinitesimal affine geometry of metric spaces endowed with a dilatation structure}, author={Marius Buliga}, journal={arXiv: Metric Geometry}, year={2008} }

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible…

## 20 Citations

### Deformations of normed groupoids and differential calculus. First part

- Mathematics
- 2009

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is…

### A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections

- Mathematics
- 2008

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and…

### Sub-riemannian geometry from intrinsic viewpoint

- Mathematics
- 2012

Gromov proposed to extract the (di erential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath eodory distance. One of the most striking features of a regular…

### Braided spaces with dilations and sub-riemannian symmetric spaces

- Mathematics
- 2010

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian…

### Uniform refinements, topological derivative and a differentiation theorem in metric spaces

- Mathematics
- 2009

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger…

### Emergent algebras as generalizations of differentiable algebras, with applications

- Mathematics
- 2009

We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Algebraically, irqs are related with…

### Maps of metric spaces

- Mathematics
- 2011

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive…

### COLIN implies LIN for emergent algebras

- Mathematics
- 2021

Emergent algebras, first time introduced in arXiv:0907.1520, are families of quasigroup operations indexed by a commutative group, which satisfy some algebraic relations and also topological…

### Metric Lie groups admitting dilations

- MathematicsArkiv för Matematik
- 2021

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$,…

### The em-convex rewrite system

- MathematicsArXiv
- 2018

We introduce and study em (or "emergent"), a lambda calculus style rewrite system inspired from dilations structures in metric geometry. Then we add a new axiom (convex) and explore its consequences.…

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