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

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

2005-10-31

arxiv.org/abs/math/0511004v2

Journal of Differential Geometry, 82(1), 115-154, 2009

Mathematics

Group Theory

44 pages, 12 figures, to appear in the Journal of Differential Geometry

Scientific paper

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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