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arxiv: 2606.09702 · v1 · pith:QKTD2JPWnew · submitted 2026-06-08 · 🧮 math.GR · math.GT

On profinite rigidity, Grothendieck pairs, and the second homology of some 3-orbifold groups

Carl-Fredrik Nyberg-Brodda This is my paper

Pith reviewed 2026-06-27 14:44 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords profinite rigidityGrothendieck pairssecond homology3-orbifold groupsWeeks manifoldhyperbolic 3-manifoldsquaternion algebra
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The pith

Lattices between the Weeks manifold fundamental group and its normalizer in PSL(2,C) are absolutely profinitely rigid, with some yielding Grothendieck pairs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an effective method to compute second homology for 3-orbifold groups that arise as finite extensions of the fundamental group of a hyperbolic rational homology 3-sphere. It applies the method to all cocompact lattices between π1 of the Weeks manifold and its normalizer in PSL(2,C), establishing their absolute profinite rigidity and completing earlier work. The computations also show that the second homology of the normalizer group Γ_O is Z/2Z while identifying other lattices whose derived subgroups have vanishing second homology, which then produce Grothendieck pairs.

Core claim

The lattices between π1(W) and its normalizer in PSL2(C) are absolutely profinitely rigid. Moreover H2(Γ_O, Z) ≅ Z/2Z, and there exist lattices whose derived subgroup is Γ_O^1 with vanishing second homology, yielding Grothendieck pairs. The method computes these homologies explicitly for the relevant orbifold groups and also for some finite extensions of Fibonacci manifold groups.

What carries the argument

An effective algorithm for computing H2 of orbifold groups arising as finite extensions of the fundamental group of hyperbolic rational homology 3-spheres, which makes the general theoretical computability practical in this case.

If this is right

  • All cocompact lattices between π1(W) and its normalizer are absolutely profinitely rigid.
  • H2(Γ_O, Z) is isomorphic to Z/2Z.
  • Certain lattices with derived subgroup Γ_O^1 have vanishing second homology and therefore yield Grothendieck pairs.
  • The method also computes second homology for finite extensions by orientable isometries of Fibonacci manifold fundamental groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The effective H2 method may apply to other hyperbolic 3-orbifold groups satisfying the same extension hypothesis.
  • The combination of rigidity and controlled homology could be used to produce further examples of groups with identical profinite completions.

Load-bearing premise

The orbifold groups in question arise as finite extensions of the fundamental group of a hyperbolic rational homology 3-sphere.

What would settle it

An explicit computation or construction showing that H2 of one of the lattices between π1(W) and its normalizer differs from the reported value, or that one such lattice shares its profinite completion with a non-isomorphic group.

Figures

Figures reproduced from arXiv: 2606.09702 by Carl-Fredrik Nyberg-Brodda.

Figure 1
Figure 1. Figure 1: The subgroup lattice of the full group Isom(W) ∼= D6 of isometries of the Weeks manifold, cf. [MV98, [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The first and second page of the homological spectral se￾quence for the extension G = ΓO of the Weeks manifold W by its full isometry group Isom(W) ∼= D6, showing that H1(ΓO, Z) = Z2⊕Z2, and H2(ΓO, Z) = Z2. The data for all morphisms drawn out is contained in the matrices M2 and M3 of Tables 2 and 3. denote the n-fold cyclic covering of S 3 branched over the figure-eight knot. Then Mn is a hyperbolic manif… view at source ↗
read the original abstract

The second homology group is of central importance in the study of profinite rigidity of $3$-manifold groups. Although general and deep results imply that the integral homology of cocompact hyperbolic $3$-orbifold groups is computable in principle, the resulting algorithm is not practical. We develop an effective method for computing $H_2$ in the case of orbifold groups arising as finite extensions of the fundamental group of hyperbolic rational homology $3$-spheres. As a special case, this yields explicit computations of the second homology groups of all cocompact lattices between $\pi_1(\mathcal{W})$ and its normalizer in $\mathrm{PSL}_2(\mathbb C)$, where $\mathcal{W}$ is the Weeks manifold. We also show that these lattices are absolutely profinitely rigid, completing work by Bridson, McReynolds, Reid & Spitler in this setting. As a special case, we determine that $H_2(\Gamma_{\mathcal{O}}, \mathbb Z) \cong \mathbb{Z} / 2\mathbb{Z}$, where $\Gamma_{\mathcal{O}}$ is the normalizer of the group of units $\Gamma_{\mathcal{O}}^1$ in a choice of maximal order $\mathcal{O}$ of the quaternion algebra associated to $\mathcal{W}$, thereby answering a question of Bridson & Reid. Although this non-vanishing obstructs one possible construction of Grothendieck pairs in $\Gamma_{\mathcal{O}}^1 \times \Gamma_{\mathcal{O}}^1$, we use our computations to show the vanishing of the second homology of another lattice whose derived subgroup is $\Gamma_{\mathcal{O}}^1$, which then yields Grothendieck pairs in this direct product by a theorem of Bridson & Reid. Finally, to showcase the generality of the techniques, we also compute the second homology of some finite extensions by orientable isometries of the fundamental group of some Fibonacci manifolds $M_n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops an effective algorithm for computing the second homology H_2(Γ, ℤ) of 3-orbifold groups Γ that are finite extensions of the fundamental group of a hyperbolic rational homology 3-sphere. It applies the algorithm to all cocompact lattices between π_1(W) and its normalizer in PSL_2(C), where W is the Weeks manifold, proving they are absolutely profinitely rigid. It computes H_2(Γ_O, ℤ) ≅ ℤ/2ℤ for the normalizer Γ_O of the unit group in a maximal order, answering a question of Bridson & Reid. Using a lattice with vanishing H_2 whose derived subgroup is Γ_O^1, it constructs Grothendieck pairs in the direct product. The method is also used to compute H_2 for some finite extensions of Fibonacci manifold groups.

Significance. If the algorithm and its applications are correct, the paper makes a significant contribution by providing a practical computational tool for homology in this class of groups, where general methods are not effective. It completes the profinite rigidity results for the Weeks lattices from prior work by Bridson et al., resolves an open question on the homology of Γ_O, and gives new examples of Grothendieck pairs via the Bridson-Reid theorem. The explicit computations and the generality shown with Fibonacci manifolds strengthen the case for the method's utility in the field of profinite rigidity of 3-manifold and orbifold groups.

minor comments (3)
  1. [Abstract] The abstract refers to computations for 'some Fibonacci manifolds M_n' without indicating the range of n or the number of cases treated; adding a short clause would clarify the scope of the generality claim.
  2. [Introduction (method paragraph)] The effective algorithm is described as relying on the finite-extension hypothesis for rational homology sphere orbifolds; a brief remark on why this hypothesis makes the computation practical (e.g., reduction to a finite check) would help readers follow the logic without consulting the cited prior works.
  3. [Preliminaries] Notation for the various lattices (e.g., Γ_O versus Γ_O^1) is introduced in the abstract and used throughout; a small table or diagram in the preliminaries section summarizing the inclusion relations and derived-subgroup relations would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on new effective algorithm and external theorems

full rationale

The paper develops an original effective algorithm for computing H2(Γ, Z) specifically for orbifold groups that are finite extensions of fundamental groups of hyperbolic rational homology 3-spheres. It then applies this algorithm to concrete lattices (Weeks normalizer and Fibonacci extensions) to obtain explicit values such as H2(Γ_O, Z) ≅ Z/2Z. Rigidity statements complete prior external work by Bridson–McReynolds–Reid–Spitler, and Grothendieck-pair constructions invoke a theorem of Bridson & Reid. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the paper's own inputs; all central claims rest on independently verifiable computations and cited external results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within established frameworks of group homology, profinite completions, and 3-orbifold topology; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard theorems on the homology of groups and on profinite rigidity of hyperbolic 3-manifold groups
    The method and rigidity conclusions rest on previously established results in the field.

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