{"paper":{"title":"On the large-scale geometry of diffeomorphism groups of $1$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Michael P. Cohen","submitted_at":"2016-06-13T15:33:17Z","abstract_excerpt":"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 $\\Diff_+^k(M^1)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03994","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}