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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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