**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.

- diams.pdf (
*520K*)

38 pages, 12 figures.

[Pictures by Riley]