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

Abstract

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 $1\le k\le \infty$) 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 $\underset{f\in A}{\sup }\underset{x\in M^1}{\sup }|\log f^{\prime }(x)|<\infty$ and $\underset{f\in A}{\sup }\underset{x\in M^1}{\sup }|f^{(j)}(x)|<\infty$ for every integer $2\le j\le k$. 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 $C[0,1]$.

arXiv comment17 pages, 1 figure