Problem 2.2 — Does the converse hold in the above theorem without the assumption that G is free?

Definitions and mathematical statement

The problem asks whether the converse of the preceding theorem remains true after dropping its assumption that $G$ is free. That theorem is absent, so its other hypotheses and the exact implication being reversed cannot be recovered uniquely. The surrounding programme concerns word hyperbolicity of surface-group extensions. In the customary notation, such an extension has the form \[1\longrightarrow\pi_1(S)\longrightarrow\Gamma_G\longrightarrow G\longrightarrow1,\] with the action/extension data specified by the source; this excerpt does not give their construction or a particular property of $G$ to substitute for the missing theorem.

The stated goal is to extend the Bestvina--Feighn combination principle from actions on trees to actions on Gromov-hyperbolic cell complexes, with sufficient additional structure to decide word hyperbolicity of $\Gamma_G$ in the intended setting. A finitely generated group is word hyperbolic when a Cayley graph has uniformly thin geodesic triangles. An action on a hyperbolic complex by itself is not the asserted criterion; the missing hypotheses must specify how stabilizers, attachment maps and extension geometry are controlled.

The source specifically points to the action of $\Gamma_G$ on a Rips complex of $G$. Given a word metric on $G$ and scale $R$, that complex has vertices $G$ and a simplex on each finite subset of diameter at most $R$, with the translation action of $G$; the quotient map $\Gamma_G\to G$ induces the indicated action. The kernel acts trivially in this model, so no proper action or immediate geometric-action criterion is inferred. The metric, scale, hyperbolicity and finite-generation assumptions on $G$ are not stated in the extract.

The source says there were no available examples on which to test its Problem 2.2, and announces related proposals in Section 4. This is a historical limitation of that investigation, not a current nonexistence assertion. Section 4 and the preceding theorem are not supplied. In particular, a familiar convex-cocompactness or other surface-subgroup theorem is not silently chosen as the missing converse.

Short English statement

Extend the omitted surface-extension converse beyond free quotient groups, using a combination theorem for actions on hyperbolic cell complexes rather than only trees. The exact theorem and its remaining hypotheses are missing.

Sources

• Local UnsolvedMath registry, permanent ID $11000230$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 2.2, PDF page 269; running chapter title Problems in the geometry of surface group extensions: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Bestvina--Feighn's combination theorem is named without a full citation; the preceding theorem and Section 4 proposals are absent.