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

TL;DR

This paper constructs a Sierpiński carpet with a quasisymmetric co-Hopfian property, explores its implications for hyperbolic spaces, and characterizes its quasisymmetry group, advancing understanding in geometric group theory.

Contribution

It introduces a new quasisymmetric co-Hopfian property, provides an explicit example of such a Sierpiński carpet, and describes its quasisymmetry group completely.

Findings

Constructed a Sierpiński carpet with the co-Hopfian property.

Established a quasi-isometrically co-Hopfian hyperbolic space with a Sierpiński carpet boundary.

Proved the quasisymmetry group is uncountable and equals the bi-Lipschitz group.

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 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.