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Dehn functions and Hölder extensions in asymptotic cones

  • Alexander Lytchak EMAIL logo , Stefan Wenger and Robert Young

Abstract

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

Award Identifier / Grant number: DMS-1612061

Award Identifier / Grant number: 153599

Award Identifier / Grant number: 165848

Award Identifier / Grant number: SPP 2026

Funding statement: Alexander Lytchak was partially supported by DFG grant SPP 2026. Stefan Wenger was partially supported by Swiss National Science Foundation Grants 153599 and 165848. Robert Young was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada, by a Sloan Research Fellowship, and by NSF grant DMS-1612061.

Acknowledgements

We thank the anonymous referee for useful comments.

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Received: 2016-11-22
Revised: 2018-12-05
Published Online: 2019-01-20
Published in Print: 2020-06-01

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