Home On the large-scale geometry of diffeomorphism groups of 1-manifolds
Article
Licensed
Unlicensed Requires Authentication

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

[1] S. Banach, Théorie des opérations linéaires, IMPAN Monogr. Mat. 1, Instytut Matematyczny PAN, Warsaw, 1932. Search in Google Scholar

[2] Y. de Cornulier and P. de la Harpe, Metric Geometry of Locally Compact Groups, EMS Tracts Math. 25, European Mathematical Society, Zürich, 2016. 10.4171/166Search in Google Scholar

[3] M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucia, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, Springer, New York, 2001. 10.1007/978-1-4757-3480-5Search in Google Scholar

[4] N. J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut. 21 (2008), 7–60. 10.5209/rev_REMA.2008.v21.n1.16426Search in Google Scholar

[5] M. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embedddings into Banach Spaces, De Gruyter Stud. Math. 49, De Gruyter, Berlin, 2013. 10.1515/9783110264012Search in Google Scholar

[6] C. Rosendal, A topological version of the Bergman property, Forum Math. 21 (2009), no. 2, 299–332. 10.1515/FORUM.2009.014Search in Google Scholar

[7] C. Rosendal, Large scale geometry of automorphism groups, preprint (2014), https://arxiv.org/abs/1403.3107. Search in Google Scholar

[8] C. Rosendal, Coarse geometry of topological groups, preprint (2015). 10.1017/9781108903547Search in Google Scholar

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
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2016-0149/html
Scroll to top button