
Extrinsic versus intrinsic diameter for Riemannian fillingdiscs 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 1skeleton of D or extrinsically in the
Cayley graph of G. We construct the first examples of closed manifolds
M and finitely presented groups G = π_{1}(M) for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different
mindiameter filling functions.
 Talk Slides (pdf), 268KB, Asymptotic and Probabilistic Methods in Geometric Group Theory, Geneva, June 2005

