Boundaries of Groups and Kleinian Groups — Problem 42

Definitions and mathematical statement

An embedding $f:(Z,d)\to(Z,d)$ is quasisymmetric if there is a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that for all distinct points $x,a,b\in Z$, \[ \frac{d(fx,fa)}{d(fx,fb)} \leq\eta\!\left(\frac{d(x,a)}{d(x,b)}\right). \] The map is a homeomorphism onto its image, which is not assumed to be the whole space. A compact metric space is quasisymmetrically co-Hopfian if every such self-embedding is surjective.

Determine which metric fractals are quasisymmetrically co-Hopfian, particularly boundaries of word-hyperbolic groups equipped with visual metrics. A visual metric on $\partial G$ is comparable to $\exp(-\varepsilon(\xi|\zeta)_o)$, where the Gromov product measures the common initial travel of rays from a basepoint and $\varepsilon>0$ is an admissible parameter.

The source highlights boundaries homeomorphic to Sierpiński carpets and Menger compacta as test families. These topological descriptions do not determine their quasisymmetric geometry. The task is a classification by geometric hypotheses and examples; it does not assert that every fractal or every hyperbolic boundary must be co-Hopfian.

An injective homomorphism of the underlying group is not required, and an arbitrary topological self-embedding does not suffice to disprove the property. A counterexample must give a proper image while controlling all triples of distances by one distortion function. Results for special rigid carpet families do not settle the general classification.

Short English statement

Which hyperbolic-group boundaries and other metric fractals admit no proper quasisymmetric copy of themselves?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200042 (AMR-061-0042), 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 42, PDF page 13. Original source indexed by AMR-061-0042. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.