Question 3.3 — LetG∼=π1(S2g)→ Γg be the injection given by Theorem 3.1.

Definitions and mathematical statement

Let $\rho:G\hookrightarrow\Gamma_g$, with $G\cong\pi_1(S_{2g})$, be the specific injection of the source's Theorem 3.1. The theorem, allowed $g$, surface notation and ambient group definition are absent. The source identifies $\widetilde S_{2g}\cong\mathbb H^2$, so its surface-group boundary $\partial_\infty G$ is the circle at infinity.

Question 3.3 asks whether there exists a continuous map \[ F:\partial_\infty G\longrightarrow\mathrm{PML}_0(\Sigma) \] which is $G$-equivariant: \[ F(\gamma\xi)=\rho(\gamma)F(\xi) \quad(\gamma\in G,\ \xi\in\partial_\infty G). \] The subspace $\mathrm{PML}_0(\Sigma)$, its subscript, topology and action are not defined. It must not be replaced by all projective measured laminations without source evidence. Only continuity and equivariance are requested; no injectivity, surjectivity or extension of a specified orbit map is added.

The following Section 3.5 gives separability context. In its standard ambient-group formulation, $\Gamma$ is $H$-separable when \[ \forall x\in\Gamma\setminus H\ \exists K\le\Gamma:\quad [\Gamma:K]<\infty,\ H\le K,\ x\notin K. \] The source prints $g\in G\setminus H$ after naming the ambient group $\Gamma$; the displayed definition makes that symbol mismatch explicit rather than importing the earlier surface subgroup as the ambient domain. LERF requires separability for every finitely generated $H$; GFERF requires it for every geometrically finite subgroup in a specified geometric setting.

The source motivates separability by asking whether an essential surface in a finite-volume hyperbolic $3$-manifold lifts to an embedding in a finite cover. It says geometrically finite subgroup separability often suffices for applications. It further reports that $\Gamma_g$ is not LERF for $g\ge2$, since it contains $F_2\times F_2$, while $\Gamma_1$ is LERF because it is virtually free. These are contextual source claims, not the answer to the boundary-map question. The passage ends before the next proposed separability question, whose statement is not reconstructed.

Short English statement

For the particular embedded surface group, ask for a continuous equivariant map from its boundary circle to the source's specified lamination subspace. The embedding theorem and target definition are missing. The subsequent LERF/GFERF discussion is preserved as separate context, including its ambient-symbol issue.

Sources

• Local UnsolvedMath registry, permanent ID $11000246$, exact source code catalogue entry, Question 3.3, PDF page 286. No individual proposer or proposed year is supplied. The statement refers to Theorem 3.1 and then enters Section 3.5.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page. The extract includes an A. W. Reid page footer, page 280.
• Internal references [1] and [36] for low-dimensional subgroup-separability literature; full entries absent. No complete separate citation for Theorem 3.1 is supplied.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; the research summary says only "Literature status: Truncated."
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.