Inventiones mathematicae

, Volume 180, Issue 2, pp 361–388

A Sierpiński carpet with the co-Hopfian property


DOI: 10.1007/s00222-010-0231-5

Cite this article as:
Merenkov, S. Invent. math. (2010) 180: 361. doi:10.1007/s00222-010-0231-5


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.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA