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

The group $${\text {Diff}}({\mathcal {M}})$$ of diffeomorphisms of a closed manifold $${\mathcal {M}}$$ 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 $$sp\le \dim {\mathcal {M}}$$ 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 $$(s-1)p\ge \dim {\mathcal {M}}$$ , and that for spheres the diameter is finite when $$(s-1)p<1$$ . In particular, this gives a full characterization of the diameter of $${\text {Diff}}(S^1)$$ . In addition, we show that for $${\text {Diff}}_c({\mathbb {R}}^n)$$ , if the diameter is not zero, it is infinite.