Boundaries of Groups and Kleinian Groups — Problem 52

Definitions and mathematical statement

A finitely generated group $G$ is word-hyperbolic if its Cayley graph has uniformly thin geodesic triangles. For an integer $n\geq2$, a homeomorphism of the round sphere $S^n$ is quasiconformal if, in conformal coordinate charts, it belongs locally to $W^{1,n}$ and has bounded distortion \[ \lVert Df\rVert^n\leq K\,|\det Df| \] almost everywhere. The constant $K$ can be chosen independent of the charts.

Does every word-hyperbolic group admit a faithful action on some sphere $S^n$ by quasiconformal homeomorphisms with one common finite distortion bound for all group elements, and with discrete image in $\operatorname{Homeo}(S^n)$? More explicitly, seek an injective homomorphism \[ \rho:G\longrightarrow\operatorname{Homeo}(S^n),\qquad \sup_{g\in G}K(\rho(g))<\infty, \] whose image has the identity isolated in the topology of uniform convergence of maps and their inverses.

The dimension and uniform distortion bound may depend on $G$. Uniformity is essential: it is not enough that each group element be quasiconformal with unrelated constants that grow without bound. Discreteness of the action also does not mean a properly discontinuous action on the entire compact sphere; limit sets can prevent that. The question asks for an action on a full sphere, not just the standard convergence action on the group's own possibly nonspherical Gromov boundary.

Short English statement

Can every hyperbolic group act discretely on some sphere with a uniform quasiconformal distortion bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200052 (AMR-061-0052), 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 52, PDF page 15. Original source indexed by AMR-061-0052. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.