Boundaries of Groups and Kleinian Groups — Problem 64

Definitions and mathematical statement

The source fragment, attributed to Cannon--Thurston, asks: "Is this action conjugate to a conformal action?" The antecedent specifying the group, space, and action is not included.

In a two-sphere interpretation suggested by the supplied summary, let $\rho:G\to\operatorname{Homeo}(X)$ be the intended action with $X\cong S^2$. The proposed conformal-conjugacy conclusion would be existence of a homeomorphism $h:X\to S^2$ such that \[ h\rho(g)h^{-1}\in\operatorname{Conf}(S^2)\qquad\text{for every }g\in G. \] Here the target has its round conformal structure; orientation-preserving conformal maps are Möbius maps, and allowing orientation reversal requires the corresponding anticonformal maps. No smoothness of the conjugating homeomorphism is implied by topological conjugacy.

The summary relates the fragment to Cannon's conjecture for word-hyperbolic groups with two-sphere boundary, asking for a geometric action on hyperbolic three-space. That is a contextual possible specialization, not a recovery of the missing antecedent. The source fragment itself does not state hyperbolicity, a boundary identification, discreteness, cocompactness, faithfulness, or orientation assumptions. Therefore no equivalence with a precise Cannon-conjecture formulation is asserted without the preceding source context.

Short English statement

Can an unspecified Cannon--Thurston action be topologically conjugated to a conformal action? The supplied summary suggests a two-sphere boundary setting, but the original action and hypotheses are missing.

Sources

• Local UnsolvedMath registry, permanent ID 6200064, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 64"; proposed year 2005, no proposer in metadata, source fragment attributed to Cannon--Thurston. Registry status partially solved, classification PARTIAL-PROGRESS.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 64, PDF page 17, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. The antecedent of "this action" is missing; no PDF was accessed here.
• The supplied research summary relates the question to Cannon's conjecture and calls it open/partial. It mentions Markovic (2013/2015) and describes some claims as disputed/withdrawn, without identifying a paper or supplying evidence for that characterization. That editorial assertion is unverified here and is not used as a mathematical or bibliographic premise.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the document is publicly author-hosted.