Dehn functions and Hölder extensions in asymptotic cones

Alexander Lytchak, Stefan Wenger, Robert Young

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalJournal fur die Reine und Angewandte Mathematik
DOIs
StateAccepted/In press - Jan 1 2019

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Dehn Function
Asymptotic Cone
Cones
Quadratic Function
Quasi-isometry
Group theory
Closed curve
Group Theory
Invariance
Metric space
Regularity
Generalise
Invariant
Geometry

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Dehn functions and Hölder extensions in asymptotic cones. / Lytchak, Alexander; Wenger, Stefan; Young, Robert.

In: Journal fur die Reine und Angewandte Mathematik, 01.01.2019.

Research output: Contribution to journalArticle

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