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Extrinsic versus intrinsic diameter for riemannian filling–discs and van kampen diagrams

Martin Bridson Mathematical Institute, Timothy Riley University Walk

Differential Geometry mathscidoc:1609.10114

Journal of Differential Geometry, 82, (1), 115-154, 2009
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold M may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram  filling a word that represents the identity in a finitely presented group 􀀀 can either be measured intrinsically in the 1–skeleton of  or extrinsically in the Cayley graph of 􀀀. We construct the first examples of closed manifolds M and finitely presented groups 􀀀 = π1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min–diameter filling functions.
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@inproceedings{martin2009extrinsic,
  title={EXTRINSIC VERSUS INTRINSIC DIAMETER FOR RIEMANNIAN FILLING–DISCS AND VAN KAMPEN DIAGRAMS},
  author={Martin Bridson, and Timothy Riley},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911182803623000771},
  booktitle={Journal of Differential Geometry},
  volume={82},
  number={1},
  pages={115-154},
  year={2009},
}
Martin Bridson, and Timothy Riley. EXTRINSIC VERSUS INTRINSIC DIAMETER FOR RIEMANNIAN FILLING–DISCS AND VAN KAMPEN DIAGRAMS. 2009. Vol. 82. In Journal of Differential Geometry. pp.115-154. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911182803623000771.
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