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

Martin Bridson and Tim Riley

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 D filling a word that represents the identity in a finitely presented group G can either be measured intrinsically in the 1-skeleton of D or extrinsically in the Cayley graph of G. We construct the first examples of closed manifolds M and finitely presented groups G = π1(M) for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min-diameter filling functions.

  • Talk Slides (pdf), 268KB, Asymptotic and Probabilistic Methods in Geometric Group Theory, Geneva, June 2005