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- W2443347568 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 $1leq kleqinfty$) of a compact $1$-manifold $M^1$ ($=I$ or $mathbb{S}^1$). We characterize the relative property (OB) in such groups: $AsubseteqDiff_+^k(M^1)$ has property (OB) relative to $Diff_+^k(M^1)$ if and only if $displaystylesup_{fin A}sup_{xin M^1}|log f'(x)|<infty$ and $displaystylesup_{fin A}sup_{xin M^1}|f^{(j)}(x)|<infty$ for every integer $2leq jleq k$. 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 $C[0,1]$." @default.
- W2443347568 created "2016-06-24" @default.
- W2443347568 creator A5037723105 @default.
- W2443347568 date "2016-06-13" @default.
- W2443347568 modified "2023-09-27" @default.
- W2443347568 title "On the large-scale geometry of diffeomorphism groups of $1$-manifolds" @default.
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