Some examples of quasiisometries of nilpotent Lie groups
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Xiangdong Xie
Abstract
We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N, we construct quasiisometries from N to itself that is not at finite distance from any map that is a composition of left translations and automorphisms. We also construct bi-Lipschitz maps of the Heisenberg groups that send vertical lines to curves that are not vertical lines.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS–1265735
Funding statement: Partially supported by NSF grant DMS–1265735.
Acknowledgements
The examples were discovered while I was visiting MSRI in the fall of 2011. I would like to express my gratitude to MSRI for its hospitality. I also would like to thank Bruce Kleiner for useful discussions.
References
[1] Z. Balogh, R. Hoefer-Isenegger and J. Tyson, Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 621–651. 10.1017/S0143385705000593Search in Google Scholar
[2] A. Dranishnikov, S. Ferry and S. Weinberger, Large Riemannian manifolds which are flexible, Ann. of Math. (2) 157 (2003), no. 3, 919–938. 10.4007/annals.2003.157.919Search in Google Scholar
[3] A. Eskin, D. Fisher and K. Whyte, Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs, Ann. of Math. (2) 176 (2012), no. 1, 221–260. 10.4007/annals.2012.176.1.3Search in Google Scholar
[4] A. Eskin, D. Fisher and K. Whyte, Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups, Ann. of Math. (2) 177 (2013), no. 3, 869–910. 10.4007/annals.2013.177.3.2Search in Google Scholar
[5] B. Farb and L. Mosher, On the asymptotic geometry of abelian-by-cyclic groups, Acta Math. 184 (2000), no. 2, 145–202. 10.1007/BF02392628Search in Google Scholar
[6] Y. Guivarc’h, Croissance polynomiale et periodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 353–379. 10.24033/bsmf.1764Search in Google Scholar
[7] R. Karidi, Geometry of balls in nilpotent Lie groups, Duke Math. J. 74 (1994), no. 2, 301–317. 10.1215/S0012-7094-94-07415-2Search in Google Scholar
[8] A. Koranyi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309–338. 10.1007/BF01388609Search in Google Scholar
[9] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60. 10.2307/1971484Search in Google Scholar
[10] N. Shanmugalingam and X. Xie, A rigidity property of some negatively curved solvable lie groups, preprint (2009), https://arxiv.org/abs/1001.0150. 10.4171/CMH/269Search in Google Scholar
[11] X. Xie, Quasisymmetric maps on the boundary of a negatively curved solvable Lie group, Math. Ann. 353 (2012), no. 3, 727–746. 10.1007/s00208-011-0700-1Search in Google Scholar
[12] X. Xie, Quasisymmetric maps on reducible Carnot groups, Pacific J. Math. 265 (2013), no. 1, 113–122. 10.2140/pjm.2013.265.113Search in Google Scholar
[13]
X. Xie,
Large scale geometry of negatively curved
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Tight contact structures on the Brieskorn spheres -Σ(2,3,6n-1) and contact invariants
- Some examples of quasiisometries of nilpotent Lie groups
- Spans of special cycles of codimension less than 5
- The braided Thompson's groups are of type F∞
- Fonctions régulues
- A Dixmier--Douady theory for strongly self-absorbing C*-algebras
- Monodromy of A-hypergeometric functions
- Primitive ideals, twisting functors and star actions for classical Lie superalgebras
Articles in the same Issue
- Frontmatter
- Tight contact structures on the Brieskorn spheres -Σ(2,3,6n-1) and contact invariants
- Some examples of quasiisometries of nilpotent Lie groups
- Spans of special cycles of codimension less than 5
- The braided Thompson's groups are of type F∞
- Fonctions régulues
- A Dixmier--Douady theory for strongly self-absorbing C*-algebras
- Monodromy of A-hypergeometric functions
- Primitive ideals, twisting functors and star actions for classical Lie superalgebras
