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Spaces with almost Euclidean Dehn function

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We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are \({\mathrm{CAT}}(0)\). This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper \({\mathrm{CAT}}(0)\) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being \({\mathrm{CAT}}(0)\) up to that scale.

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Acknowledgements

I wish to thank the anonymous referee for very useful comments which have led to several improvements.

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  1. Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700, Fribourg, Switzerland

    Stefan Wenger

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  1. Stefan Wenger
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Correspondence to Stefan Wenger.

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Communicated by Andreas Thom.

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This research was supported by Swiss National Science Foundation Grants 153599 and 165848.

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Wenger, S. Spaces with almost Euclidean Dehn function. Math. Ann. 373, 1177–1210 (2019). https://doi.org/10.1007/s00208-019-01819-2

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