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Extrinsic versus intrinsic diameter for Riemannian filling-discs and van Kampen diagrams

M. R. Bridson·T. R. Riley
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Abstract

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold 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 its 1-skeleton or extrinsically in the Cayley graph of the group. We construct the first examples of closed manifolds and finitely presented groups for which this choice -- intrinsic versus extrinsic -- gives rise to qualitatively different min-diameter filling functions.

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M. R. Bridson, T. R. Riley. 2009-06-27. Extrinsic versus intrinsic diameter for Riemannian filling-discs and van Kampen diagrams. https://arxiv.org/abs/math/0511004

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