Boundaries of Groups and Kleinian Groups — Problem 50

Definitions and mathematical statement

The initial request, whether a prior construction can use smaller $m$, specifically $m=n+1$, has no antecedent in the extracted record and cannot be formalized independently. The text also says that the same problem applies to complex hyperbolic space.

The explicit Kapovich subproblem concerns a standard inclusion of hyperbolic symmetric spaces $W_n\hookrightarrow W_{n+1}$ and its boundary map \[ i:X=\partial_\infty W_n\hookrightarrow Y=\partial_\infty W_{n+1}. \] The extracted symbol "HHn" is not defined. If it denotes quaternionic hyperbolic space $\mathbb H_{\mathbb H}^{n}$, that choice and its admissible dimension range must be confirmed; a real- or complex-hyperbolic interpretation is a different problem. The boundaries are equipped with the intended visual metric structures.

A quasisymmetric embedding $f:X\to Y$ is an embedding onto its image with a distortion homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that \[ \frac{d_Y(f(x),f(a))}{d_Y(f(x),f(b))} \le\eta\left(\frac{d_X(x,a)}{d_X(x,b)}\right) \] for all distinct $x,a,b\in X$. Local rigidity asks whether every such $f$ sufficiently close to $i$ has the form \[ f=(\partial_\infty g)\circ i \qquad\text{for some }g\in\operatorname{Isom}(W_{n+1}). \] The equality concerns the parametrized embedding, not merely congruence of its image. Since $X\ne Y$, the source's identity means the inclusion $i$.

For example, in the compact-open boundary topology a precise version would require some $\varepsilon>0$ such that \[ \sup_{x\in X}d_Y(f(x),i(x))<\varepsilon \quad\Longrightarrow\quad f=(\partial_\infty g)\circ i. \] The source does not specify this topology, metric normalization, or whether a fixed quasisymmetry control $\eta$ is required. The displayed neighborhood is therefore a candidate convention, not an added verified hypothesis. The supplied research summary calls the specific boundary rigidity statement open and mentions related higher-rank embedding rigidity without establishing this conclusion.

Short English statement

Is the standard boundary inclusion locally rigid among quasisymmetric embeddings, so every sufficiently nearby embedding is induced by an ambient isometry? The hyperbolic-space notation, closeness topology, and missing smaller-dimension antecedent need source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $6200050$, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 50." Proposer David Fisher, 2005; explicit subproblem attributed to Misha Kapovich.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 50, recorded PDF page 15; https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Recorded access 2026-07-29; pdf-text-normalized extraction.
• Supplied OPEN-TRIAGE research summary mentions Kleiner--Leeb and Eskin--Fisher--Whyte as related rigidity literature, without complete references or a theorem resolving this boundary statement.