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A Sierpiński carpet with the co-Hopfian property

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Abstract

Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasi-isometrically co-Hopfian Gromov hyperbolic space with a Sierpiński carpet boundary at infinity. In addition, we give a complete description of the quasisymmetry group of the constructed Sierpiński carpet. This group is uncountable and coincides with the group of bi-Lipschitz transformations.

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References

  1. Ahlfors, L.: Lectures on Quasiconformal Mappings. Van Nostrand Mathematical Studies, vol. 10. Van Nostrand, Princeton (1966). Manuscript prepared with the assistance of Clifford J. Earle, Jr.

    MATH  Google Scholar 

  2. Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150(1), 127–183 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2), 266–306 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourdon, M., Pajot, H.: Rigidity of quasi-isometries for some hyperbolic buildings. Comment. Math. Helv. 75(4), 701–736 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buyalo, S., Schroeder, V.: Elements of Asymptotic Geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2007)

    Book  MATH  Google Scholar 

  6. David, G., Semmes, S.: Fractured Fractals and Broken Dreams. Self-similar Geometry through Metric and Measure. Oxford Lecture Series in Mathematics and Its Applications, vol. 7. Clarendon/Oxford University Press, New York (1997)

    MATH  Google Scholar 

  7. Farb, B., Handel, M.: Commensurations of Out(F n ). Publ. Math. Inst. Hautes Études Sci. 105, 1–48 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Ghys, E., de la Harpe, P. (eds.): Sur les Groupes Hyperboliques d’Après Mikhael Gromov. Progress in Mathematics, vol. 83. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  9. Gromov, M.: Hyperbolic Groups. Essays in Group Theory. Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer, New York (1987)

    Google Scholar 

  10. de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2000)

    MATH  Google Scholar 

  11. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    MATH  Google Scholar 

  12. Kapovich, I., Benakli, N.: Boundaries of hyperbolic groups. In: Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001). Contemp. Math., vol. 296, pp. 39–93. Am. Math. Soc., Providence (2002)

    Google Scholar 

  13. Kapovich, I., Wise, D.T.: On the failure of the co-Hopf property for subgroups of word-hyperbolic groups. Isr. J. Math. 122, 125–147 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kapovich, M., Kleiner, B.: Coarse Alexander duality and duality groups. J. Differ. Geom. 69(2), 279–352 (2005)

    MATH  MathSciNet  Google Scholar 

  15. Kapovich, M., Kleiner, B.: Hyperbolic groups with low-dimensional boundary. Ann. Sci. Éc. Norm. Super. (4) 33(5), 647–669 (2000)

    MATH  MathSciNet  Google Scholar 

  16. Sela, Z.: Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II. Geom. Funct. Anal. 7(3), 561–593 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tyson, J.: Quasiconformality and quasisymmetry in metric measure spaces. Ann. Acad. Sci. Fenn. Math. 23(2), 525–548 (1998)

    MathSciNet  Google Scholar 

  18. Whyburn, G.T.: Topological characterization of the Sierpiński curve. Fundam. Math. 45, 320–324 (1958)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Sergei Merenkov

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  1. Sergei Merenkov
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Correspondence to Sergei Merenkov.

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Supported by NSF grant DMS-0653439.

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Merenkov, S. A Sierpiński carpet with the co-Hopfian property. Invent. math. 180, 361–388 (2010). https://doi.org/10.1007/s00222-010-0231-5

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  • DOI: https://doi.org/10.1007/s00222-010-0231-5

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