Kirby Problem 3.8

Definitions and mathematical statement

For a group $G$, its profinite completion is the topological group \[ \widehat G=\varprojlim_{N\triangleleft G,\ [G:N]<\infty}G/N, \] where each finite quotient is discrete and the inverse limit carries its compact inverse-limit topology. An isomorphism of profinite completions here is an isomorphism of topological groups. This invariant records all finite quotients together with their compatibility maps.

Let $M_1$ and $M_2$ be connected complete finite-volume hyperbolic three-manifolds, with sectional curvature $-1$. Suppose \[ \widehat{\pi_1(M_1)}\cong\widehat{\pi_1(M_2)}. \] Must there be an isometry $M_1\to M_2$? Closed and cusped manifolds are both included. Orientation reversal is allowed in the conclusion: the question concerns their underlying hyperbolic manifolds rather than marked oriented ones.

No identification of the fundamental groups is part of the hypothesis. The issue is whether finite quotient information alone forces that geometric identification. Likewise, the question compares manifolds within the hyperbolic class; it does not assert that every finitely generated group with the same completion must itself be one of these manifold groups.

Short English statement

Does the complete collection of finite quotients of a hyperbolic three-manifold group determine the manifold up to isometry?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2806 (KP-3.8), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.8. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.