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

Martin Bridson and Tim Riley

Preprint, October 2005.

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 \tilde{M} or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram D filling a word that represents 1 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=\pi_1 M for which this choice -- intrinsic versus extrinsic -- gives rise to qualitatively different min-diameter filling functions.

38 pages, 12 figures.

[Pictures by Riley]