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

  • Michael P. Cohen EMAIL logo
Published/Copyright: April 8, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 1

Abstract

We apply the framework of Rosendal to study the large-scale geometry of the topological groups Diff+k⁡(M1), consisting of orientation-preserving Ck-diffeomorphisms (for 1≤k≤∞) of a compact 1-manifold M1 (=I or 𝕊1). We characterize the relative property (OB) in such groups: A⊆Diff+k⁡(M1) has property (OB) relative to Diff+k⁡(M1) if and only if supf∈Asupx∈M1|logf′(x)|<∞ and supf∈A⁡supx∈M1⁡|f(j)⁢(x)|<∞ for every integer j with 2≤j≤k. We deduce that Diff+k⁡(M1) has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if k<∞. We show that the groups Diff+1⁡(I) and Diff+1⁡(𝕊1) are quasi-isometric to the infinite-dimensional Banach space C⁢[0,1].

Keywords: Quasi-isometry; property (OB); diffeomorphism group
MSC 2010: 20F65; 22F05

Communicated by Manfred Droste

Acknowledgements

I would like to thank C. Rosendal, B. Sari, and A. Akhmedov for helpful discussions and remarks.

References

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Received: 2016-06-29
Revised: 2016-12-30
Published Online: 2017-04-08
Published in Print: 2018-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Logarithmic Sobolev inequalities for Moebius measures on spheres
  3. Delta sets for nonsymmetric numerical semigroups with embedding dimension three
  4. Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L2(ℝ+;dx/x)
  5. Remarks on Lp-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems
  6. On blocks with one modular character
  7. On the large-scale geometry of diffeomorphism groups of 1-manifolds
  8. Lp boundedness of rough bi-parameter Fourier integral operators
  9. Coclosed G2-structures inducing nilsolitons
  10. Existence of solutions for a semirelativistic Hartree equation with unbounded potentials
  11. On two questions concerning representations distinguished by the Galois involution
  12. Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains
  13. Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds
  14. The splitting of cohomology of p-groups with rank 2
  15. Star-group identities on units of group algebras: The non-torsion case
  16. Expansion for cubes in the Heisenberg group
  17. Semilinear Robin problems resonant at both zero and infinity
  18. Purely infinite simple Kumjian–Pask algebras
https://doi.org/10.1515/forum-2016-0149
Keywords for this article
Quasi-isometry; property (OB); diffeomorphism group

Articles in the same Issue

  1. Frontmatter
  2. Logarithmic Sobolev inequalities for Moebius measures on spheres
  3. Delta sets for nonsymmetric numerical semigroups with embedding dimension three
  4. Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L2(ℝ+;dx/x)
  5. Remarks on Lp-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems
  6. On blocks with one modular character
  7. On the large-scale geometry of diffeomorphism groups of 1-manifolds
  8. Lp boundedness of rough bi-parameter Fourier integral operators
  9. Coclosed G2-structures inducing nilsolitons
  10. Existence of solutions for a semirelativistic Hartree equation with unbounded potentials
  11. On two questions concerning representations distinguished by the Galois involution
  12. Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains
  13. Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds
  14. The splitting of cohomology of p-groups with rank 2
  15. Star-group identities on units of group algebras: The non-torsion case
  16. Expansion for cubes in the Heisenberg group
  17. Semilinear Robin problems resonant at both zero and infinity
  18. Purely infinite simple Kumjian–Pask algebras
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