On the large-scale geometry of diffeomorphism groups of $1$-manifolds

We apply the framework of Rosendal to study the large-scale geometry of the topological groups $\Diff_+^k(M^1)$, consisting of orientation-preserving $C^k$-diffeomorphisms (for $1\leq k\leq\infty$) of a compact $1$-manifold $M^1$ ($=I$ or $\mathbb{S}^1$). We characterize the relative property (OB) in such groups: $A\subseteq\Diff_+^k(M^1)$ has property (OB) relative to $\Diff_+^k(M^1)$ if and only if $\displaystyle\sup_{f\in A}\sup_{x\in M^1}|\log f'(x)|<\infty$ and $\displaystyle\sup_{f\in A}\sup_{x\in M^1}|f^{(j)}(x)|<\infty$ for every integer $2\leq j\leq k$. We deduce that $\Diff_+^k(M^1)$ has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if $k<\infty$. We show that the groups $\Diff_+^1(I)$ and $\Diff_+^1(\mathbb{S}^1)$ are quasi-isometric to the infinite-dimensional Banach space $C[0,1]$.