Question 3.4 — Is Γg GFERF for g≥ 2?

Definitions and mathematical statement

A subgroup $H\leq G$ is separable if, for every $x\in G\setminus H$, there exist a finite group $Q$ and a homomorphism $q:G\to Q$ with $q(x)\notin q(H)$. Equivalently, $H$ is closed in the profinite topology, or is the intersection of the finite-index subgroups of $G$ containing it.

Question 3.4 asks whether $\Gamma_g$ is GFERF for every $g\geq2$. In the usual expansion of this term, GFERF means that every geometrically finite subgroup is separable. Thus, if $\mathcal G_g$ denotes the chapter's geometrically finite subgroup class, the question is whether \[ \forall g\geq2\quad\forall H\in\mathcal G_g\quad \forall x\in\Gamma_g\setminus H\quad \exists q:\Gamma_g\to Q\text{ finite}:\quad q(x)\notin q(H). \] Neither $\Gamma_g$ nor geometrical finiteness is defined in this excerpt. The mapping-class interpretation is suggested by the volume context, but no specific convex-cocompact, relative or other geometric subgroup class can be substituted silently. GFERF is not automatically LERF, which asks separability of every finitely generated subgroup.

The registry suggests a negative general answer from nonseparable geometric subgroups, while acknowledging variable meanings and incomplete verification. Without an exact eligible subgroup class and cited counterexample, that suggestion does not resolve this source-specific assertion.

Short English statement

For each $g\geq2$, are all subgroups deemed geometrically finite in the source separable in $\Gamma_g$? The ambient group and geometric-finiteness convention are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000247$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Question 3.4, PDF page 287: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• No definition of $\Gamma_g$ or GFERF and no primary counterexample or theorem are supplied. The registry's negative suggestion is explicitly incompletely verified.