## A metric characterization of Carnot groups

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## Abstract:

We give a short axiomatic introduction to Carnot groups and their subRiemannian and subFinsler geometry. We explain how such spaces can be metrically described as exactly those proper geodesic spaces that admit dilations and are isometrically homogeneous.## References

- André Bellaïche,
*The tangent space in sub-Riemannian geometry*, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. MR**1421822**, DOI 10.1007/978-3-0348-9210-0_{1} - V. N. Berestovskiĭ,
*Homogeneous manifolds with an intrinsic metric. I*, Sibirsk. Mat. Zh.**29**(1988), no. 6, 17–29 (Russian); English transl., Siberian Math. J.**29**(1988), no. 6, 887–897 (1989). MR**985283**, DOI 10.1007/BF00972413 - V. N. Berestovskiĭ,
*Similarly homogeneous locally complete spaces with an intrinsic metric*, Izv. Vyssh. Uchebn. Zaved. Mat.**11**(2004), 3–22 (Russian); English transl., Russian Math. (Iz. VUZ)**48**(2004), no. 11, 1–19 (2005). MR**2179443** - Sergei Buyalo and Viktor Schroeder,
*Moebius characterization of the boundary at infinity of rank one symmetric spaces*, preprint (2012). - Marius Buliga,
*A characterization of sub-Riemannian spaces as length dilation structures constructed via coherent projections*, Commun. Math. Anal.**11**(2011), no. 2, 70–111. MR**2780883** - Cornelia Drutu and Michael Kapovich,
*Lectures on geometric group theory*, Manuscript (2011). - David M. Freeman,
*Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations*, Indiana Univ. Math. J.**62**(2013), no. 1, 311–331. MR**3158510**, DOI 10.1512/iumj.2013.62.4872 - Andrew M. Gleason,
*Groups without small subgroups*, Ann. of Math. (2)**56**(1952), 193–212. MR**49203**, DOI 10.2307/1969795 - Mikhael Gromov,
*Carnot-Carathéodory spaces seen from within*, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR**1421823** - Enrico Le Donne,
*Geodesic manifolds with a transitive subset of smooth biLipschitz maps*, Groups Geom. Dyn.**5**(2011), no. 3, 567–602. MR**2813528**, DOI 10.4171/GGD/140 - Enrico Le Donne,
*Metric spaces with unique tangents*, Ann. Acad. Sci. Fenn. Math.**36**(2011), no. 2, 683–694. MR**2865538**, DOI 10.5186/aasfm.2011.3636 - G. A. Margulis and G. D. Mostow,
*The differential of a quasi-conformal mapping of a Carnot-Carathéodory space*, Geom. Funct. Anal.**5**(1995), no. 2, 402–433. MR**1334873**, DOI 10.1007/BF01895673 - G. A. Margulis and G. D. Mostow,
*Some remarks on the definition of tangent cones in a Carnot-Carathéodory space*, J. Anal. Math.**80**(2000), 299–317. MR**1771529**, DOI 10.1007/BF02791539 - Richard Montgomery,
*A tour of subriemannian geometries, their geodesics and applications*, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR**1867362**, DOI 10.1090/surv/091 - Deane Montgomery and Leo Zippin,
*Topological transformation groups*, Robert E. Krieger Publishing Co., Huntington, N.Y., 1974. Reprint of the 1955 original. MR**0379739**

## Additional Information

**Enrico Le Donne**- Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland
- MR Author ID: 867590
- Email: enrico.e.ledonne@jyu.fi
- Received by editor(s): April 18, 2013
- Received by editor(s) in revised form: April 28, 2013
- Published electronically: September 18, 2014
- Additional Notes: The author thanks IPAM and all of the people involved in the program, ‘Interactions Between Analysis and Geometry’, during which there was the opportunity to discuss these results
- Communicated by: Jeremy Tyson
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**143**(2015), 845-849 - MSC (2010): Primary 53C17, 53C60, 22E25, 58D19
- DOI: https://doi.org/10.1090/S0002-9939-2014-12244-1
- MathSciNet review: 3283670