Is the fundamental group of every orientable complete finite-volume hyperbolic 3-manifold determined, up to isomorphism, among all finitely generated residually finite groups by its set of finite quotients — i.e., is it absolutely profinitely rigid? Recent work (through August 2026, most recently in preprint form) has settled the relative versions: all lattices in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) are profinitely rigid among themselves and among all finitely generated virtually 3-manifold groups, and a finite list of arithmetic lattices is absolutely rigid. The absolute conjecture, however, remains open, with a precise surviving core: excluding a finitely generated residually finite group that is not virtually a 3-manifold group but has the same profinite completion as the fundamental group of a finite-volume hyperbolic 3-manifold.

Background

The profinite completion Γ^\widehat{\Gamma} of a group Γ\Gamma is the inverse limit of its finite quotient groups; equivalently, Γ^\widehat{\Gamma} packages the set of isomorphism classes of finite quotients of Γ\Gamma. A group is residually finite if finite quotients separate its elements, and a finitely generated group Γ\Gamma is said to be profinitely rigid among a class A\mathcal{A} if every Δ∈A\Delta\in\mathcal{A} with Δ^≅Γ^\widehat{\Delta}\cong\widehat{\Gamma} satisfies Δ≅Γ\Delta\cong\Gamma; rigidity among all finitely generated residually finite groups is called absolute. By Mostow–Prasad rigidity, a complete finite-volume hyperbolic 3-manifold MM (constant sectional curvature −1-1, finitely many ends called cusps) is determined up to isometry by its fundamental group, which is a torsion-free lattice in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) (a discrete subgroup of finite covolume); such groups are linear and hence residually finite. Asking whether the finite quotients of π1(M)\pi_1(M) determine MM is therefore a finite-quotient analogue of Mostow rigidity, and it is the hyperbolic endgame of the broader program of detecting 3-manifolds from finite quotients, a program motivated in part by Agol's virtual fibering theorem (every closed hyperbolic 3-manifold has a finite cover that fibers over the circle). The non-hyperbolic part of the program is essentially settled. Wilkes (2016) gave computable criteria classifying graph manifolds (3-manifolds whose JSJ decomposition — the canonical splitting along essential tori into Seifert-fibered pieces, i.e. pieces foliated by disjoint circles, and hyperbolic pieces — has only Seifert-fibered pieces) by the profinite completions of their groups, within the class of 3-manifold groups (fundamental groups of 3-manifolds); Yi Liu (2020) proved that among finitely generated 3-manifold groups the profinite completion determines π1(M)\pi_1(M) up to finite ambiguity together with the Thurston norm (a norm on second homology measuring the minimal complexity of embedded surfaces representing a homology class) and fibering data; and profinite determination genuinely fails outside the hyperbolic class: Funar (2013) constructed non-isomorphic Sol torus-bundle groups (fundamental groups of mapping tori of torus automorphisms, carrying Sol, one of Thurston's eight model geometries) with isomorphic profinite completions, and Xiaoyu Xu (2024) showed that the profinite completion of a mixed 3-manifold (one with both hyperbolic and Seifert-fibered JSJ pieces) need not determine its Seifert part unless peripheral structure (the data of the cusp subgroups inside the completion) is included. On the hyperbolic side, absolute rigidity is known only sporadically: Bridson–McReynolds–Reid–Spitler (Annals 2020) proved that specific arithmetic Kleinian groups (arithmetic discrete subgroups of PSL(2,C)\mathrm{PSL}(2,\mathbb{C})) — the Bianchi group PSL(2,Z[ω])\mathrm{PSL}(2,\mathbb{Z}[\omega]) with ω\omega a primitive third root of unity, the non-uniform lattice of minimal covolume, and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume) — are absolutely profinitely rigid; Cheetham-West (2022) gave the first closed fibered hyperbolic examples, including the first non-orientable one; Bridson–Reid–Spitler (2023) treated direct products of rigid 3-manifold groups. Many further families are known to be rigid relative to the class of compact 3-manifold groups (Bridson–Reid's figure-eight knot complement; Xu's Dehn-filling families — Dehn filling attaches a solid torus to a toral boundary component along a slope — including the Whitehead link complement; Huang's geometric-convergence families; Cheetham-West–Xu's two-bridge links (links admitting a diagram with exactly two local maxima) and 3-tangle Montesinos links (links built by gluing rational tangles)). The frontier moved in August 2026, when Xiaoyu Xu proved that all lattices in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) are profinitely rigid among themselves with Out(Γ^)≅Out(Γ)\mathrm{Out}(\widehat{\Gamma})\cong\mathrm{Out}(\Gamma), and that any finitely generated group virtually a lattice in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) (virtually = containing a finite-index subgroup with the stated property) is profinitely rigid among all finitely generated virtually 3-manifold groups. Xu's paper explicitly presents this as a step toward the conjecture of Bridson and Reid that lattices in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) are profinitely rigid in the absolute sense — the conjecture that remains open and whose torsion-free (manifold) case is the subject of this problem. Standard background on 3-manifold groups and their finite-quotient properties is in Aschenbrenner–Friedl–Wilton's survey and Tianwei Liu's 2025 survey.

Problem Statement

Let MM be an orientable complete finite-volume hyperbolic 3-manifold, so π1(M)\pi_1(M) is a torsion-free lattice in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}). Is it the case that for every such MM, π1(M)\pi_1(M) is profinitely rigid in the absolute sense — that is, if Δ\Delta is any finitely generated residually finite group with Δ^≅π1(M)^\widehat{\Delta}\cong\widehat{\pi_1(M)} (equivalently, with the same set of isomorphism classes of finite quotient groups), must Δ≅π1(M)\Delta\cong\pi_1(M)? This is the torsion-free (manifold) case of the conjecture of Bridson and Reid that every lattice in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) is profinitely rigid in the absolute sense. An affirmative answer must cover all orientable complete finite-volume hyperbolic 3-manifolds, closed and cusped; a negative answer must exhibit, for some such MM, a finitely generated residually finite group Δ≇π1(M)\Delta\not\cong\pi_1(M) with Δ^≅π1(M)^\widehat{\Delta}\cong\widehat{\pi_1(M)}, together with proofs of residual finiteness, of the profinite isomorphism, and of non-isomorphism. As of August 2026 the relative cases appear settled — in recent preprints, π1(M)\pi_1(M) is distinguished among lattices in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) and among all finitely generated virtually 3-manifold groups — so the remaining open core is the case where Δ\Delta is a finitely generated residually finite group not virtually isomorphic to a 3-manifold group; the problem asks for this remaining case in full, without restricting the target class, the manifold class, or the answer form.

References

  1. Tianwei Liu, 'Toward the profinite rigidity of hyperbolic 3-manifolds', 2025, arXiv:2508.20110, DOI 10.48550/arXiv.2508.20110, https://arxiv.org/abs/2508.20110
  2. M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler, 'Absolute profinite rigidity and hyperbolic geometry', 2020, Annals of Mathematics 192, DOI 10.4007/annals.2020.192.3.1 (preprint arXiv:1811.04394, 2018), https://arxiv.org/abs/1811.04394
  3. Xiaoyu Xu, 'Profinite rigidity in lattices of PSL(2,C)', 2026, arXiv:2608.07350, DOI 10.48550/arXiv.2608.07350, https://arxiv.org/abs/2608.07350
  4. Tamunonye Cheetham-West, Xiaoyu Xu, 'Profinite rigidity for two-bridge links and 3-tangle Montesinos links', 2026, arXiv:2603.20822, DOI 10.48550/arXiv.2603.20822, https://arxiv.org/abs/2603.20822
  5. Xiaoyu Xu, 'Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds', 2024, arXiv:2412.05229, DOI 10.48550/arXiv.2412.05229, https://arxiv.org/abs/2412.05229
  6. Xiaoyu Xu, 'Profinite almost rigidity in 3-manifolds', 2024, arXiv:2410.16002, DOI 10.48550/arXiv.2410.16002, https://arxiv.org/abs/2410.16002
  7. Tamunonye Cheetham-West, 'Absolute profinite rigidity of some closed fibered hyperbolic 3-manifolds', 2022, arXiv:2205.08693, DOI 10.48550/arXiv.2205.08693, https://arxiv.org/abs/2205.08693
  8. M. R. Bridson, A. W. Reid, R. Spitler, 'Absolute profinite rigidity, direct products, and finite presentability', 2023, arXiv:2312.06058, DOI 10.48550/arXiv.2312.06058, https://arxiv.org/abs/2312.06058
  9. Yu Huang, 'Profinite rigidity and geometric convergence', 2025, arXiv:2501.02234, DOI 10.48550/arXiv.2501.02234, https://arxiv.org/abs/2501.02234
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  17. Louis Funar (with an appendix by Louis Funar and Andrei Rapinchuk), 'Torus bundles not distinguished by TQFT invariants', 2013, Geometry & Topology 17, 2289–2344, DOI 10.2140/gt.2013.17.2289, https://msp.org/gt/2013/17-4/p09.xhtml (constructs non-isomorphic Sol torus-bundle 3-manifold groups with isomorphic profinite completions)