A Sierpi\'nski carpet with the co-Hopfian property

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.