On the large-scale geometry of diffeomorphism groups of $1$-manifolds
Michael P. Cohen

TL;DR
This paper studies the large-scale geometry of groups of smooth, orientation-preserving diffeomorphisms of 1-manifolds, characterizing their properties and showing their quasi-isometric relations to Banach spaces.
Contribution
It characterizes the relative property (OB) in diffeomorphism groups and establishes their quasi-isometry to Banach spaces for finite smoothness levels.
Findings
Groups have property (OB) iff derivatives are uniformly bounded.
Diff groups are quasi-isometric to $C[0,1]$ for $k=1$.
Finite smoothness implies a non-trivial quasi-isometry class.
Abstract
We apply the framework of Rosendal to study the large-scale geometry of the topological groups , consisting of orientation-preserving -diffeomorphisms (for ) of a compact -manifold ( or ). We characterize the relative property (OB) in such groups: has property (OB) relative to if and only if and for every integer . We deduce that has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if . We show that the groups and are quasi-isometric to the infinite-dimensional Banach space .
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