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

Metric Geometry