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Infinite groups from the profinite point of view

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Pith's one-line read This survey argues that for finitely generated residually finite groups, the profinite completion determines only a short list of properties, while almost every other structural feature is invisible to the finite quotients.

desk verdict A reliable survey of profinite rigidity, indispensable for newcomers, with minor proofreading slips and a legitimate self-citation load. read the letter →

arxiv 2506.08755 v1 pith:AGQMJ54L submitted 2025-06-10 math.GR

Holger Kammeyer , Steffen Kionke This is my paper
classification math.GR MSC 20E1820E2620F6520G3022E40
keywords profinitecompletionrigidityS-arithmeticgroupsbranchGrothendieckpairsℓ2-Bettinumberscongruencesubgrouppropertyuniformamenability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This survey examines how much of a finitely generated residually finite group is determined by its set of finite quotients, which is the same information as its profinite completion. The paper's central thesis is that profinite properties are rare: bounded cohomology, higher $\ell^2$-Betti numbers, property (T), the fixed-point property FA, finiteness properties, the centre, and torsion-freeness all fail to be profinite, with explicit non-isomorphic groups sharing a profinite completion for each. The properties known to be profinite are short and specific: satisfying a law, polynomial word growth, uniform amenability, the first $\ell^2$-Betti number among groups satisfying approximation in degree one, and, within arithmetic classes with finite congruence kernel, the sign of the Euler characteristic and the covolume. The survey therefore gives a map of what finite quotients can and cannot reveal about an infinite group, together with the open problems, including the profinite rigidity of non-abelian free groups and of $\mathrm{SL}_n(\mathbb{Z})$ for $n \ge 3$.

What carries the argument

The profinite completion $\widehat{\Gamma} = \varprojlim \Gamma/N$ is the inverse limit of all finite quotients, and two finitely generated residually finite groups have the same finite quotients exactly when their completions are isomorphic. Negative examples come from two sources: the congruence subgroup property, which makes $\widehat{\Gamma}$ a product of local factors so that local isomorphisms between number fields or algebraic groups can be traded without changing the completion; and branch-group and telescope constructions, where the completion is an iterated (semi)direct product depending only on low-level data, giving Grothendieck pairs. Positive results use word maps, whose continuity forces laws to lift from $\Gamma$ to $\widehat{\Gamma}$; the approximation formula expressing the first $\ell^2$-Betti number as a limit of finite-index Betti numbers; and Euler-characteristic and covolume computations that are controlled by the congruence subgroup property. The load-bearing named objects are the congruence kernel, the adelic superrigidity principle, and the iterated wreath product.

What would settle it

Compute the profinite completion of $\mathrm{SL}_3(\mathbb{Z}[1/p])$ for a prime $p$ and compare it with the product of the local groups $\mathrm{SL}_3(\mathbb{Z}_\ell)$; a finite quotient distinguishing the two would falsify the finite-congruence-kernel description behind the arithmetic rigidity results. A differently targeted check: find two finitely generated residually finite groups with isomorphic profinite completions where exactly one has polynomial word growth, which would refute the profiniteness of laws and polynomial growth.

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Extended reading notes

Core claim

The central claim, on the paper's own terms, is that the profinite completion of a finitely generated residually finite group keeps only a thin algebraic and analytic skeleton of the group. The negative half is carried by two construction principles: for $S$-arithmetic groups with finite congruence kernel, the completion factorises over local places, so locally isomorphic quadratic forms or number fields can be exchanged to produce non-isomorphic groups with the same completion; and for branch groups built by spinal actions, the completion is an iterated wreath product depending only on first-level data, yielding $2^{\aleph_0}$ Grothendieck pairs, i.e. inclusions $\Delta \subsetneq \Gamma$ of a proper subgroup that induce an isomorphism of profinite completions. In particular, one member of such a pair can be amenable while the other contains a non-abelian free group, so amenability is not profinite. The positive half lists exactly what does survive: laws, polynomial growth, uniform amenability, the first $\ell^2$-Betti number under approximation in degree one, and, among arithmetic groups with the congruence subgroup property, the sign of the Euler characteristic and equality of covolumes.

Load-bearing premise

The load-bearing premise is the conjecture that every higher-rank $S$-arithmetic subgroup of a simple algebraic group has finite congruence kernel; if this fails for any such group, the profinite completions used in the solitary-group and lattice classifications no longer have the advertised product structure and those theorems would need revision.

Editorial extensions

If this is right

  • Laws transfer to the profinite completion and back, so any property definable by a law, including being abelian, nilpotent, solvable, or virtually such, is profinite.
  • Polynomial word growth is profinite, so a finitely generated residually finite group has polynomial growth exactly when its profinite completion satisfies the corresponding law-theoretic condition.
  • Uniform amenability is profinite even though amenability is not; the branch-group constructions yield $2^{\aleph_0}$ Grothendieck pairs in which the proper subgroup is amenable and the ambient group contains a non-abelian free subgroup.
  • The first $\ell^2$-Betti number is determined by the profinite completion for all groups satisfying approximation in degree one, hence in particular for all finitely presented groups.
  • Among $S$-arithmetic groups with finite congruence kernel, profinite commensurability preserves the sign of the Euler characteristic and, within a fixed Lie group, the covolume; consequently finite-volume hyperbolic 3-manifolds with profinitely isomorphic fundamental groups would have equal volume if the relevant arithmetic cases are representative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The survey leaves implicit that the congruence subgroup property is the single structural bottleneck: if it were proved in full, the exception lists in the solitary-lattice theorems would shrink, and the corresponding classifications would become unconditional rather than conditional.
  • A testable extension of the branch-group method is to use the telescope construction on other iterated semidirect products, potentially producing 2-generated groups with prescribed profinite completions and thereby turning many finitely generated profinite groups into profinite completions of finitely generated discrete groups.
  • The listed profinite properties are all semidecidable from an enumeration of the finite quotients, while the non-profinite properties are not; a natural sharpening of the survey's dichotomy is to ask which profinite properties are recursively computable from the finite quotient data, a question the paper does not address.
  • Given the survey's evidence that torsion and homology are invisible to the completion, the open problem of whether the first $\ell^2$-Betti number is profinite for all finitely generated residually finite groups is likely to have a negative answer; the non-approximable torsion groups with positive first $\ell^2$-Betti number are the natural place to look for a counterexample.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper is a survey of recent work on the question which properties of finitely generated residually finite groups are determined by their profinite completion. It focuses on S-arithmetic groups and branch groups. The main thesis is that profinite properties are rare: the survey collects many constructions of non-isomorphic groups with isomorphic profinite completions (spinor groups, Chevalley groups, branch groups yielding uncountably many Grothendieck pairs) and identifies the few properties that are profinite (laws, polynomial growth, uniform amenability, the first ℓ2-Betti number among Lücky groups, and the sign of the Euler characteristic for S-arithmetic groups with the congruence subgroup property). The paper also states several open problems. The exposition includes proofs of standard results and clearly marks which quoted theorems are conditional on Serre's congruence subgroup conjecture.

Significance. The survey provides a valuable and current map of the field. Its main claims are supported by a coherent body of examples and by the explicit distinction between unconditional results and those depending on Serre's conjecture. The paper is particularly strong in highlighting the contrast between higher-rank S-arithmetic and branch-group constructions, and in listing the known profinite properties. A minor caveat is that a large fraction of the surveyed theorems are the authors' own results, which is natural in a survey of this area but means the selection is closely tied to their research program. Overall, if the manuscript is polished, it will be a useful reference for graduate students and researchers.

minor comments (4)
  1. [§5.1] In the proof of Lemma 5.1, the displayed chain 'bH2 ∼= H2 ∼= H1 ∼= bH2' is garbled; it should state that the profinite completions of H1 and H2 are isomorphic, for example 'bH2 ≅ Ψ(H1) ≅ bH1'.
  2. [§3.2] The notation 'cΓn' in the discussion of the groups Γn± is missing a hat; it should be \widehat{\Gamma_n^\pm}.
  3. [§2.1] In Lemma 2.4, the displayed isomorphism 'bΓab ∼= [(Γab)' is typeset incorrectly; it should read \widehat{\Gamma^{ab}} \cong (\widehat{\Gamma})^{ab}.
  4. [§3.10] The phrase 'not even a profinte invariant' contains a typo; it should be 'profinite'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey reports known results with explicit attributions; its central claim does not reduce to its own inputs.

full rationale

This is a survey, not an original derivation. Theorems and constructions are quoted from published sources, including several prior papers by the authors, but the survey does not rederive them and it openly credits them (e.g., Theorems 3.15, 3.18, 3.19, and 4.9 are cited to [28], [30], [29], and [37]). The central negative examples, such as the profinitely isomorphic spinor groups Spin(q7,2)(Z) and Spin(q3,6)(Z), are supported by the stated congruence subgroup property for spinor groups of Witt index at least two and by concrete quadratic form isometries, not by a fitted parameter or by a self-referential definition. Where results depend on Serre's conjecture on the congruence subgroup property, the paper explicitly marks the conditionality, for instance by saying that certain exceptions 'can again be dropped if Serre’s conjecture on CSP is true.' No equation is shown to be equivalent to its own input by construction, and no fitted quantity is relabeled as a prediction. The proof of Lemma 5.1 contains a garbled isomorphism chain, but the intended argument is the standard open-subgroup correspondence and is not circular. Self-citations are numerous but they are used as acknowledgments of prior theorems, not as the sole justification of a claim that the survey is supposed to establish independently. Therefore the derivation chain contains no circular step and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The survey introduces no free parameters and no invented entities. Its content rests on standard background theorems plus the explicitly conditional use of Serre's conjecture on the congruence subgroup property.

assumptions (4)
  • domain assumption Serre's conjecture on the congruence subgroup property: higher-rank S-arithmetic groups have finite congruence kernel.
    Invoked conditionally in Theorems 3.12, 3.14, 3.15, and 3.17 (Sections 3.8 and 3.9). The survey repeatedly notes that exceptions disappear 'if Serre's conjecture on CSP is true'.
  • standard math Standard background results: classification of finite simple groups, Bass-Serre tree theory, Margulis superrigidity, Borel's ℓ2-cohomology computations, Gromov's polynomial growth theorem.
    Used throughout as accepted theorems from the literature to construct examples and prove rigidity statements.
  • standard math Lück approximation theorem for finitely presented groups (and more generally FP2 groups) in degree 1.
    Used in Section 5.2 to define Lücky groups and in the proof of Theorem 5.8.
  • standard math Ribes-Zalesskii correspondence between open subgroups of the profinite completion and finite index subgroups of the group (Lemma 2.5).
    Background for all arguments relating finite quotients to the profinite completion.

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Pith. "Pith review of Infinite groups from the profinite point of view." pith.science (2026). https://pith.science/paper/AGQMJ54L

@misc{pith2026250608755,
  author       = {Pith},
  title        = {Pith review of: Infinite groups from the profinite point of view},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGQMJ54L}},
  note         = {Machine review of arXiv:2506.08755}
}
abstract

We survey recent work ranging around the question in how far a group, or a property of a group, is determined by the set of finite quotient groups. Our focus lies on $S$-arithmetic groups, branch groups, and their relatives.

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