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

The diameter of a disc filling a loop in the universal covering of a Rie- mannian manifold M may be measured extrinsically using the distance function on f M or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagramfilling a word that represents 1 in a finitely presented group can eit her be measured intrinsically in the 1-skeleton ofor extrinsically in the Cay- ley graph of . We construct the first examples of closed manif olds M and finitely presented groups = �1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min-diameter filling functions.