Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

The group Diff (M) of diffeomorphisms of a closed manifold M is naturally equipped with various right-invariant Sobolev norms W^s,p. Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when sp≤ dim M and s< 1). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when (s- 1) p≥ dim M, and that for spheres the diameter is finite when (s- 1) p< 1. In particular, this gives a full characterization of the diameter of Diff (S¹). In addition, we show that for Diff _c(Rⁿ) , if the diameter is not zero, it is infinite.