Abstract
We establish three criteria of hyperbolicity of a graph in terms of “average width of geodesic bigons”. In particular we prove that if the ratio of the Van Kampen area of a geodesic bigon \(\beta \) and the length of \(\beta \) in the Cayley graph of a finitely presented group G is bounded above then G is hyperbolic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This means that there exists \(\varepsilon >0\) such that for every geodesic segment [x, z] and any \(y\in (x,z)\) the probability that a random path from x to z passes through y is greater than \(\varepsilon \), see [3].
References
Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential Theory—Surveys and Problems (Prague, 1987). Lecture Notes in Math. 1344, p. 123. Springer, Berlin
Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Ghys, E., Haefliger, A., Verjovsky, A. (eds.) Group Theory from a Geometric Viewpoint, pp. 64–168. World Scientific (1991)
Gekhtman, I., Gerasimov, V., Potyagailo, L., Yang, W.: Martin boundary covers Floyd boundary. Invent. Math. 223, 759–809 (2021)
Gersten, S.M.: Problems on automatic groups. In: Baumslag, G., Miller, C.F. (eds.) Algorithms and Classification in Combinatorial Group Theory, pp. 225–232. Springer, New York (1992)
Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory, pp. 75–263. Springer, Berlin (1987)
Lyndon, R.C., Shupp, P.E.: Combinatorial Group Theory. Springer, Berlin (1977)
Papasoglu, P.: Strongly automatic groups are hyperbolic. Invent. Math. 121, 323–334 (1995)
Acknowledgements
The authors are thankful to the research Grant LABEX CEMPI (ANR-11-LABX-0007-01) for providing a support to Victor Gerasimov to visit the University of Lille in Summer 2022 when the works on the project has been started. The authors are also grateful to the Brazilian-French Network in Mathematics for providing a support to Leonid Potyagailo to visit Federal University of Belo Horizonte in December 2022 where the paper has been completed. The authors are grateful to the referee for helpful comments and for pointing out inaccuracies.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
At the request of the publisher, the authors declare that they have no relevant financial or non-financial interests to disclose.
Additional information
Communicated by Mikhail Belolipetsky.
To the memory of our friend Sasha Anan’in.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and by the Brazilian-French Network in Mathematics.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gerasimov, V., Potyagailo, L. Integral criteria of hyperbolicity for graphs and groups. São Paulo J. Math. Sci. (2024). https://doi.org/10.1007/s40863-023-00396-2
Accepted:
Published:
DOI: https://doi.org/10.1007/s40863-023-00396-2