  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.

Group Theory;Differential Geometry