# Quasisymmetric embeddings of slit Sierpi\'nski carpets

## TL;DR

This paper characterizes when dyadic slit Sierpiński carpets can be quasisymmetrically embedded into the plane, linking this to a transboundary Loewner property and properties of associated pillowcase spheres.

## Contribution

It introduces the Transboundary Loewner Property (TLP) as a key criterion for embedding dyadic slit carpets into the plane and relates embeddings to the regularity of associated pillowcase spheres.

## Key findings

- Dyadic slit carpets are embeddable iff they satisfy TLP.
- Embedding is equivalent to the pillowcase sphere being quasisymmetric to the sphere.
- Associated pillowcase spheres are Ahlfors 2-regular if and only if the carpet embeds.

## Abstract

We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\'nski carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet $X$ can be associated to a "pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic to the sphere $\mathbb{S}^2$. We show that $X$ quasisymmetrically embeds into the plane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if and only if $\widehat{X}$ is Ahlfors $2$-regular.