Boundaries of Groups and Kleinian Groups — Problem 62

Definitions and mathematical statement

Let $\mathcal S\subset[0,1]^2$ be the standard Sierpiński carpet: subdivide a square into nine equal squares, remove the open central square, and repeat this operation in every retained square. Give the intersection of the retained stages its Euclidean distance $d$.

A metric $d'$ on $\mathcal S$ belongs to its quasisymmetric gauge if the identity $(\mathcal S,d)\to(\mathcal S,d')$ is a homeomorphism whose distance ratios are controlled by a fixed distortion homeomorphism. The metric $d'$ is Ahlfors $Q$-regular if it has a Borel measure $\mu$ and a constant $C\geq1$ satisfying \[ C^{-1}r^Q\leq\mu(B_{d'}(x,r))\leq Cr^Q \quad(0<r\leq\operatorname{diam}_{d'}\mathcal S). \] Determine the Ahlfors-regular conformal dimension \[ \operatorname{Cdim}_{AR}(\mathcal S)= \inf\{Q:\text{an Ahlfors $Q$-regular metric lies in this gauge}\}. \] Also determine whether this infimum is attained by such a metric. The Euclidean Hausdorff dimension $\log8/\log3$ is an upper bound, not an asserted value of the infimum.

The metric gauge is essential: replacing the carpet by any homeomorphic compactum with an arbitrary metric changes the optimization problem. Exact determination requires a matching lower bound and construction or limiting argument. Attainment is a separate issue from knowing the numerical infimum; a sequence of admissible metrics with dimensions approaching that number need not supply a metric realizing it.

Short English statement

What is the Ahlfors-regular conformal dimension of the standard carpet, and does a metric attain it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200062 (AMR-061-0062), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 62, PDF page 17. Original source indexed by AMR-061-0062. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.
• S3 Anttila and Eriksson-Bique, Cartesian products of Sierpiński carpets do not attain their conformal dimension (2026). The single-carpet attainment question remains distinct from the product result. https://arxiv.org/abs/2604.19600.