Boundaries of Groups and Kleinian Groups — Problem 77

Definitions and mathematical statement

Let $M$ be a closed hyperbolic $n$-manifold and $G=\pi_1(M)$. A group $\Gamma$ is residually finite if every $\gamma\ne1$ has nonidentity image under some homomorphism from $\Gamma$ to a finite group.

Problem (Misha Kapovich). For a short exact sequence \[ 1\longrightarrow A\longrightarrow\Gamma\longrightarrow G\longrightarrow1, \qquad A\text{ printed as ``Zp''}, \] is $\Gamma$ residually finite? The source then asks, as a reformulation, whether there is a finite-index subgroup $G'\leq G$ for which \[ \operatorname{res}:H^2(G,A)\longrightarrow H^2(G',A) \quad\text{is the zero map}. \tag{1} \] The kernel notation is damaged/ambiguous: ${\mathbb{Z}}/p{\mathbb{Z}}$, ${\mathbb{Z}}^p$, and $p$-adic integers are different possibilities, and no range for $p$ is supplied. Formula (1) must retain the intended coefficient meaning rather than silently choose one.

Under the plausible finite-cyclic interpretation $A={\mathbb{Z}}/p{\mathbb{Z}}$, an untwisted $H^2(G,A)$ describes central extensions. The short exact sequence as printed does not explicitly assume centrality or specify the conjugation action on $A$; a nontrivial action instead uses the corresponding coefficient module. These are essential qualifications of the cohomological formulation.

For a fixed central extension with class $e\in H^2(G,A)$, restriction vanishing for that extension means $\operatorname{res}(e)=0$, i.e. splitting over $G'$. The source asks the stronger-looking uniform zero-map condition for all classes. In a finite-coefficient setting its relationship to the universal question over all extensions requires the relevant finiteness and passage-to-common-subgroup argument; it is not confused with a claim about one class without those conditions. The finite versus infinite kernel distinction also matters for any residual-finiteness/splitting equivalence.

The summary reports partial progress, attributes a "$p=1$" case to Agol, and mentions poorly specified residual-finiteness terminology without a theorem reference. That $p=1$ description does not resolve the ambiguous coefficient notation, and no precise dimensional or coefficient scope is authenticated. The general problem is reported open in the supplied evidence.

Short English statement

Are the stated extensions of closed hyperbolic-manifold groups residually finite, equivalently in the intended setting can finite-index restriction kill the relevant degree-two cohomology? The kernel notation Zp, centrality/module action, and single-class versus uniform restriction quantifiers need source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 6200077, source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 77"; proposer: Misha Kapovich, proposed year 2005. Status partially solved, classification PARTIAL-PROGRESS, updated 2026-08-06.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 77, PDF page 20, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Reported PDF-text-normalized extraction and access: 2026-07-29. Current status NEEDS_REVIEW; the source labels it a problem. Rights: public author-hosted PDF; statement-reuse terms NEEDS_REVIEW.
• The summary names Agol and a $p=1$ case but gives no primary reference or precise theorem; its additional "jp'-RF"/linear-by-hyperbolic wording is not defined in the record.