On the notions of lower Ricci curvature bound for discrete graphs
Hervé Pajot1
1 Institut Fourier, UMR 5582, Laboratoire de mathématiques Université Grenoble Alpes CS 40700, 38058 Grenoble cedex 9, France
Séminaire de théorie spectrale et géométrie, Volume 36 (2019-2021), pp. 103-126

The Ricci curvature plays an important role in Riemannian geometry. The assumption that the manifold has nonnegative Ricci curvature implies some geometric and topological constraints (For instance, the diameter of the manifold is bounded and so the manifold is compact. This is the famous Bonnet–Myers Theorem). In these notes, we present several approaches to extend this kind of results in the setting of discrete graphs, in particular Cayley graphs of finitely generated groups.

Published online:
DOI: 10.5802/tsg.374

Hervé Pajot 1

1 Institut Fourier, UMR 5582, Laboratoire de mathématiques Université Grenoble Alpes CS 40700, 38058 Grenoble cedex 9, France 
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Hervé Pajot. On the notions of lower Ricci curvature bound for discrete graphs. Séminaire de théorie spectrale et géométrie, Volume 36 (2019-2021), pp. 103-126. doi: 10.5802/tsg.374

[1] C. Ané; S. Blachère; D. Chafaï; P. Fougères; I. Gentil; F. Malrieu; C. Roberto; G. Scheffer Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses, 10, Société Mathématique de France, 2000 | Zbl

[2] D. Bakry; Z. Qian Volume comparison theorems without Jacobi fields, Current trends in Potential Theory (Theta Series in Advanced Mathematics), Volume 4, The Theta Foundation, Bucarest, 2005, pp. 115-122 | MR | Zbl

[3] A. Bar-Natan; M. Duchin; R. Kropfoller Conjugaison curvature for Cayley graphs (2020) (arXiv:1712.02484, to appear in Journal of topology and analysis)

[4] M. Bridson; A. Haefliger Metric spaces with non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999 | Zbl | DOI

[5] D. Burago; Y. Burago; S. Ivanov A course in metric geometry, Graduate Texts in Mathematics, 33, American Mathematical Society, 2001 | Zbl

[6] I. Chavel Riemannian geometry: a modern introduction, 98, Cambridge Nniversity Press, 2006 | Zbl | DOI

[7] F. Chung; S. T. Yau Logarithmic Harnack inequalities, Math. Res. Lett., Volume 3 (1996), pp. 793-812 | MR | DOI | Zbl

[8] P. De La Harpe Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000 | Zbl

[9] S. Gallot; D. Hulin; J. Lafontaine Riemannian geometry, Universitext, Springer, 2004 | Zbl | DOI

[10] B. Kleiner A new proof of Gromov’s theorem for groups of polynomial growth, J. Am. Math. Soc., Volume 23 (2010), pp. 815-829 | DOI | MR | Zbl

[11] M. Ledoux The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse, Math., Volume 9 (2000) no. 2, pp. 305-366 | DOI | MR | Numdam | Zbl

[12] Y. Lin; S. T. Yau Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett., Volume 17 (2010), pp. 343-356 | DOI | MR | Zbl

[13] J. Lott; C. Villani Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., Volume 169 (2009), pp. 903-991 | DOI | MR | Zbl

[14] Y. Ollivier Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., Volume 256 (2009), pp. 810-864 | DOI | MR | Zbl

[15] Y. Ollivier; C. Villani A curved Brunn–Minkowski inequality on the discrete hypercube; Or what is the Ricci curvature of the discrete hypercube, SIAM J. Discr. Math., Volume 26 (2012), pp. 983-996 | DOI | MR | Zbl

[16] H. Pajot The discrete Bochner formula and the Ricci curvature of Cayley graphs (in preparation)

[17] H. Pajot; E. Russ Analyse dans les espaces métriques, Savoirs actuels, CNRS Editions; EDP Sciences, 2018 | Zbl

[18] L. Saloff-Coste Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, 289, Cambridge University Press, 2002 | Zbl

[19] K.-Th. Sturm On the geometry of metric measure spaces. I., Acta Math., Volume 196 (2006) no. 1, pp. 65-131 | Zbl | DOI | MR

[20] K.-Th. Sturm On the geometry of metric measure spaces. II., Acta Math., Volume 196 (2006) no. 1, pp. 133-177 | DOI | MR | Zbl

[21] N. T. Varopoulos; L. Saloff-Coste; T. Coulhon Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, 2008 (paperback reprint of the 1992 original) | Zbl

[22] C. Villani Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009 | Zbl | DOI

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