Quasi-isometric embeddings into diffeomorphism groups (Q374094)

Let \(M\) be a compact connected and oriented Riemannian manifold endowed with the volume form \(\mu\) induced by the metric, and let \(\text{Diff}(M,\mu)\) denote the group of \(C^r\)-diffeomorphisms of \(M\) acting by the identity on a neighborhood of the boundary and preserving the volume form \(\mu\). The group \(\text{Diff}(M,\mu)\) is equipped with the \(C^k\)-topology for a fixed \(k\), \(1\leq k\leq r\leq\infty\).NEWLINENEWLINEThe main focus of the paper is a study of the geometry of the identity component \(\text{Diff}_0(M,\mu)\) of the above group endowed with the right invariant \(L^p\)-metric. In this metric, the distance \(\text{d}_p{(g_{0}, g_{1})}\) between two diffeomorphisms \(g_0, g_1 \in \text{Diff}_0(M,\mu)\) is defined as the infimum of the \(L^p\)-length of a smooth isotopy in \(\text{Diff}_0(M,\mu)\) connecting \(g_0\) and \(g_1\).NEWLINENEWLINEIf \(p=2\), the group \(\text{Diff}_0(M,\mu)\) can be equipped with a Riemannian metric inducing the above \(L^2\)-length. As shown by \textit{V. I. Arnol'd} [Ann. Inst. Fourier 16, No. 1, 319--361 (1966; Zbl 0148.45301)], the geodesics of this metric are the solutions of the equations of the flow of an incompressible fluid. This makes the case \(p=2\) particularly interesting.NEWLINENEWLINEAs main results, the authors prove that for the manifold \(M\) of dimension at least two and with a specific condition satisfied by the fundamental group \(\pi_1(M)\), they can construct quasi-isometric embeddings of either free abelian groups (with word metric) or direct products of non-abelian free groups (with word metric) into the metric group \((\text{Diff}_0(M,\mu),\text{d}_p)\). With this condition on \(\pi_1(M)\), it follows in particular, that \((\text{Diff}_0(M,\mu),\text{d}_p)\) has infinite diameter.