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Residual Finiteness Growth in Two-Step Nilpotent Groups

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Pith's one-line read For a finitely generated two-step nilpotent group, the residual finiteness growth is at most $\log^{d(\varphi_{\mathbb{C}})+1}$, with the exponent read off from the complex Mal'cev completion, and for commutator subgroup rank one or two…

desk verdict Main results are new and likely correct; the broad Theorem 1.1 is missing a finite-index reduction, and two expository remarks contain errors, but the core for I-groups is solid. read the letter →

arxiv 2505.21090 v1 pith:76CXYY7F submitted 2025-05-27 math.GR

Jonas Deré , Joren Matthys This is my paper · ORCID
classification math.GR MSC 20F6920F1820E2615A22
keywords residualfinitenessgrowthtwo-stepnilpotentgroupsMal'cevcompletionmatrixpencilsskew-symmetricmatricesminoridealsHeisenberggroupoverGaussianintegerspolylogarithmic
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

Residual finiteness growth measures how large a finite quotient must be to show a non-trivial element is not the identity. This paper proves that for every finitely generated two-step nilpotent group, that growth is bounded by a power of the logarithm whose exponent is computed from an integer invariant of the commutator form: the minimum, over bases of the complexification, of the maximal rank of a complex linear combination of the defining matrices. Because this number depends only on the complex Mal'cev completion, the upper bound is an invariant of the group rather than of its generating set. For groups whose commutator subgroup has rank one or two, including the Heisenberg group over the Gaussian integers, the paper proves the bound is exact, so the growth is precisely $\log^{d+1}$. The paper conjectures exactness in full generality, which would make every such growth an odd power of $\log$.

What carries the argument

The engine is the matrix pencil $M_x=\sum_{i=1}^n x_iA_i$ of skew-symmetric integer matrices encoding $\varphi$, together with the invariant $d(\varphi_{\mathbb C})$ defined by minimizing the maximal rank of $\mathbb C$-linear combinations over bases of $\mathbb C^n$. Divisibility of central elements is expressed as a minimum of $|\operatorname{Im}_{\mathbb Z_{p^k}}\sum_i a_iA_i|\cdot p^k$ over prime powers and projections $a$, which connects separation to ranks of $\mathbb Z_p$-reductions, and the prime-density theorem supplies primes $p=O(\log\|v\|)$ for which the reduction behaves like the complexification. For the exact results, the canonical (Kronecker) normal form of a skew matrix pencil $xA_1+yA_2$\,---\,blocks $F(\alpha,k)$, $F(\infty,k)$, $S(k)$ and zero\,---\,is used to compute the largest $d$ whose variety $V(I_d(M_x))$ lies in a hyperplane, and a combinatorial tuple lemma plus a resultant identity show that $y^d$ lies in the ideal $I_d(M_x)$, exactly the membership the lower bound requires.

What would settle it

Take any explicit $I(m,2)$-group and compute, from the Kronecker normal form of its pencil $xA_1+yA_2$, the largest $d$ for which $V(I_d(M_x))$ lies in the hyperplane $y=0$; the paper's exactness claim fails if $y^d\notin I_d(M_x)$ for that $d$. Equivalently, the conjecture fails if some two-step I-group has $V(I_d(M_x))$ contained in a hyperplane but no power $(\sum v_ix_i)^d$ belonging to $I_d(M_x)$ with non-zero integer $v$.

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Extended reading notes

Core claim

The central claim is that for a two-step I-group $G_\varphi$ defined by a full alternating bilinear form $\varphi:\mathbb{Z}^m\times\mathbb{Z}^m\to\mathbb{Z}^n$ with matrices $A_1,\ldots,A_n$, the residual finiteness growth satisfies $\mathrm{RF}_{G_\varphi}(r)\preceq \log^{d(\varphi_{\mathbb{C}})+1}(r)$, where $d(\varphi_{\mathbb{C}})=\min_{\{a^{(1)},\ldots,a^{(n)}\}}\max_j \operatorname{rank}_{\mathbb{C}}\sum_i a^{(j)}_iA_i$ and the minimum runs over bases of $\mathbb{C}^n$. The proof reduces non-trivial elements to central elements $(0,v)$, detects them by reducing modulo carefully chosen primes, and uses a prime-density theorem to keep the prime as small as $O(\log\|v\|)$. The lower-bound theorem shows $\log^{d+1}\preceq\mathrm{RF}_{G_\varphi}$ whenever $(\sum_i v_ix_i)^d$ lies in the ideal generated by the $d\times d$ minors of the matrix pencil $\sum_i x_iA_i$; for commutator rank at most two this membership is proved using the canonical (Kronecker) normal form of complex skew-symmetric matrix pencils, yielding equality $\mathrm{RF}_{G_\varphi}=\log^{d(\varphi_{\mathbb{C}})+1}$ and in particular $\mathrm{RF}_{H_3(\mathbb{Z}[i])}=\log^3$.

Load-bearing premise

The exact asymptotics rest on the imported classification of complex skew-symmetric matrix pencils and on the determinant-ideal computations built on it, and the upper bound rests on the prime-density theorem that guarantees a suitable small prime $p=O(\log\|v\|)$ for every non-zero central element.

Editorial extensions

If this is right

  • For every two-step I-group, a non-trivial element of word norm at most $r$ is detected by a finite quotient of size at most $C\log^{d(\varphi_{\mathbb{C}})+1}(Cr)$, with the exponent fixed by the complex Mal'cev completion.
  • For every $I(m,1)$-group, $\mathrm{RF}_{G_\varphi}(r)=\log^{\operatorname{rank}A+1}(r)$, so every odd power $\log^{2k+1}$ with $k\geq 1$ occurs as a residual finiteness growth.
  • For every $I(m,2)$-group, in particular for $H_3(\mathbb{Z}[i])$, the upper bound is exact: $\mathrm{RF}_{G_\varphi}=\log^{d(\varphi_{\mathbb{C}})+1}$, giving $\log^3$ for the Heisenberg group over the Gaussian integers.
  • There are two-step nilpotent groups with $d(\varphi_{\mathbb{C}})=1$ but with the previous $\psi$-bound arbitrarily large, so the older general bound is not sharp.
  • If the paper's conjecture holds, every two-step nilpotent residual finiteness growth is an odd power of $\log$ depending only on the complex Mal'cev completion, which would support quasi-isometric invariance of this growth.

Reading between the lines

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

  • Editorial inference: the group-theoretic conjecture is equivalent to a commutative-algebra statement about minors of skew-symmetric matrices, so testing random integral pencils $M_x$ with $n\geq 3$ by symbolic determinant-ideal computation would either find a counterexample or increase confidence in exactness beyond rank two.
  • Editorial inference: the upper-bound strategy suggests a route to higher-step nilpotent groups, replacing the single matrix pencil by multilinear forms and seeking a prime-density analogue; the paper does not address this extension.
  • Editorial inference: if the complex-completion dependence is as strong as the paper suggests, then any two-step nilpotent groups that are quasi-isometric should have equal exponents, giving a concrete quasi-isometry invariant; the paper does not prove this direction.
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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

1 major / 4 minor

Summary. The paper studies the residual finiteness growth RF_G of two-step nilpotent groups. For an I-group G_φ defined by a full alternating bilinear form φ: Z^m × Z^m → Z^n, the authors introduce the invariant d(φ_C), computed as a min-max over bases of C^n of ranks of the complex matrices Σ a_i A_i. They prove an upper bound RF_{G_φ} ⪯ log^{d(φ_C)+1} (Theorem 3.9), construct a family of non-singular groups for which this bound is log^3 while the previously known bound ψ(G) is arbitrarily large (Section 4), prove a lower bound log^{δ+1} under an ideal-theoretic condition on minors (Theorem 5.4), and then verify that condition for commutator subgroups of rank at most 2, obtaining exact growth RF_{G_φ} = log^{d(φ_C)+1} for I(m,1)- and I(m,2)-groups (Section 6). The paper conjectures that this exactness holds in general, which would imply that the exponent in the residual finiteness growth of a two-step nilpotent group is always odd and is determined by the complex Mal'cev completion.

Significance. If correct, this is a substantial advance: it gives the first exact residual finiteness growth for nonabelian nilpotent groups beyond Heisenberg-type examples, and it ties the growth exponent to the complex Mal'cev completion in a computable way. The paper has several concrete strengths: Lemma 3.5 is an explicit closed formula for the divisibility function on the center; the upper bound uses an effective prime-density argument (Ax/van den Dries/Serre) rather than a nonconstructive bound; the lower bound is proved independently via Smith normal forms; and the rank-2 exactness rests on a detailed analysis of canonical forms for complex skew-symmetric pencils. The examples in Section 4 showing ψ(G_n) → ∞ while RF_{G_n} = log^3 are convincing and make the improvement over the previous bound quantitative. The main caveat is that the headline theorem is stated for all finitely generated infinite two-step nilpotent groups, while the proof is written only for I-groups; this gap is fixable but must be addressed.

major comments (1)
  1. [§1 (Theorem 1.1) and §3 (Theorem 3.9)] Theorem 1.1 is stated for every finitely generated infinite two-step nilpotent group, but the proof in §3, and the exactness results in §6, are carried out only for I-groups G_φ. A general finitely generated nilpotent group need not be an I-group, and the paper contains no lemma reducing the general case to the I-group case. The standard reduction is available and short: the torsion subgroup T of a finitely generated nilpotent group is finite and normal, G/T is a torsion-free I-group, and normal subgroups of G/T lift to normal subgroups of G of the same index, so that RF_G ≈ RF_{G/T}. This lemma should be stated and proved before Theorem 1.1, and the exactness theorems should be phrased with this reduction explicitly. As written, the stated generality of the main theorem is unsupported.
minor comments (4)
  1. [§6, displayed equivalence (9)] The middle assertion in the displayed equivalence (9), namely ∃l ∈ N* such that I_d(xA1+yA2) = (y^l), is false. For the S(1) block in the normal form, the pencil has rank 2 for every (x,y) ≠ (0,0), but I_2(S(1)) = (x^2, xy, y^2), which is not a principal ideal of the form (y^l). The valid equivalence is the one between the rank condition for y ≠ 0 and V(I_d) ⊂ {y = 0}; the proof of Lemma 6.13 goes through using that equivalence together with Equation (8), so the faulty middle step should be removed or corrected.
  2. [§6, Lemma 6.11 and Proposition 6.15] In Lemma 6.11 the statement says "we can construct 2d ordered tuples", but the binary-tree construction produces 2^d tuples; the same missing exponent appears in Proposition 6.15 ("2d distinct tuples" and "2d/2 tuples"). The intended meaning is clear, but the notation should be fixed.
  3. [Proof of Theorem 3.9] When Equation (5) is applied to obtain a prime p ∈ P with p ⪯ log(||v||∞) and v ∉ pZ^n, the integer m to which the effective prime theorem is applied should be specified, for example m = gcd(v_1, ..., v_n) or a nonzero coordinate of v. This is a minor clarification, since the required bound follows immediately from the stated density property.
  4. [§1, Theorem 1.1] The phrase "RF_G is smaller than log^{2k(G)+1}" should be replaced by a precise asymptotic statement such as RF_G ⪯ log^{2k(G)+1}, since the residual finiteness growth is considered up to the equivalence relation ≈ and the constant 2k(G)+1 is not a pointwise bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the invariant d(φ_C) is defined independently of RF and both the upper and lower bounds are proved from explicit algebraic and number-theoretic inputs.

full rationale

The derivation chain is self-contained and non-circular. The central invariant d(φ_C) is defined as a min-max rank of the complex matrices A_i arising from the alternating form φ_C, with no reference to residual finiteness growth. Theorem 3.9 proves the upper bound by constructing explicit congruence subgroups N_{B,D} whose index is p^{1+rank} and by choosing suitable primes using the Ax/van den Dries/Serre density theorem, quoted as Theorem 3.10. The lower bound Theorem 5.4 is proved independently from an ideal-membership hypothesis via an explicit divisibility-function estimate (Lemma 5.8), not from any fitted RF value. The exactness results for I(m,1) and I(m,2) groups rest on the external Kronecker normal form for skew matrix pencils [14,18], Smith normal form over C[t], and the combinatorial Lemma 6.11; none of these encode the conclusion. The only same-author citation, [11, Proposition 4.7], supplies an effective prime bound that is a corollary of the external density theorem rather than a restatement of the paper's main claim, so it is auxiliary and not load-bearing in a circular sense. Known benchmark values such as RF_{H_3(Z)} = log^3 are external inputs, not outputs being relabeled as predictions. The gap identified by the skeptic between I-groups and general two-step nilpotent groups is a question of scope of proof, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The ledger is clean: no fitted parameters, since d(phi_C) is a computed group invariant (a min-max over bases of C^n of matrix ranks), not a number tuned to data. All unproved inputs are standard published theorems or definitions from the literature, cited precisely. The only open item, Conjecture 2 (delta = d(phi_C) in general), is stated as a conjecture, not used as an axiom. No new entities are postulated; nothing resembling a new constant, dimension, or object is invented to make the bounds agree.

assumptions (7)
  • domain assumption Every two-step I-group is isomorphic to H_phi = (F_{m,2} x Z^n)/K_phi for a full alternating bilinear phi, via [21, Chapter 11C, Proposition 4] (Theorem 2.11).
    The whole paper works in the G_phi model; the proof of Theorem 2.11 is cited, not reproduced. Standard in the nilpotent-groups literature.
  • standard math Theorem 3.10 (Ax, van den Dries, Serre): integer polynomials with a common complex zero have a positive-density set of primes with a zero mod p; via [11, Proposition 4.7], every nonzero integer m has a prime p = O(log |m|) with p not dividing m.
    Delivers the effective primes p = O(log r) that make the upper bound polylogarithmic; quoted from [1,22,24] and [11].
  • domain assumption Theorem 6.8: every complex skew matrix pencil xA_1 + yA_2 is congruent to a block sum of F(alpha,k), F(infty,k), and S(k) blocks (Kronecker form for skew pencils), cited from [14,18].
    Load-bearing for the I(m,2) exactness computation in Lemmas 6.13 and 6.15; a published classification theorem, not proved in the paper.
  • standard math Mal'cev completion theory: exp/log correspondence for upper-triangular matrices, the Campbell-Baker-Hausdorff formula, and uniqueness of F-completions ([9,21]).
    Used to define the complex Mal'cev completion on which d(phi_C) depends; stated in Section 2.2 with citations.
  • domain assumption RF_{H_3(Z)} = log^3, the known value for the discrete Heisenberg group, used as the universal lower bound for non-abelian two-step I-groups.
    Used in Proposition 4.12, Example 3.15, and Section 5 to certify log^3 lower bounds; the paper cites [20] but gives no proof.
  • standard math Lefschetz principle for algebraically closed fields of characteristic zero (cited to [17, Theorem 3.5.5]), used to transfer statements between C and the algebraic closure of Q in Lemma 5.10.
    Justifies moving the rank-minimum computation from C to number fields in the proof that d_2 equals d_3.
  • standard math I_d(M) is invariant under congruence M -> P M Q with P, Q in GL(m,C), and the Smith Normal Form computes image sizes over Z_{p^k} (cited to [19]; used in Lemmas 5.8, 6.13, 6.15).
    Underpins the determinantal-ideal computations and the lower-bound estimate; standard linear algebra.

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Pith. "Pith review of Residual Finiteness Growth in Two-Step Nilpotent Groups." pith.science (2026). https://pith.science/paper/76CXYY7F

@misc{pith2026250521090,
  author       = {Pith},
  title        = {Pith review of: Residual Finiteness Growth in Two-Step Nilpotent Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76CXYY7F}},
  note         = {Machine review of arXiv:2505.21090}
}
abstract

Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if $g\in G$ is a non-trivial element with $\|g\|_G \leq r$, so $g$ can be written as a product of at most $r$ generators or their inverses, then we can find a homomorphism $\phi: G \to Q$ with $\phi(g) \neq e_Q$ and $|Q| \leq \text{RF}_G(r)$. The residual finiteness growth is defined as the smallest function with this property. This function has been bounded from above and below for several classes of groups, including virtually abelian, nilpotent, linear and free groups. However, for many of these groups, the exact asymptotics of $\text{RF}_G$ are unknown (in particular this is the case for a general nilpotent group), nor whether it is a quasi-isometric invariant for certain classes of groups. In this paper, we make a first step in giving an affirmative answer to the latter question for $2$-step nilpotent groups, by improving the polylogarithmic upper bound known in literature, and to show that it only depends on the complex Mal'cev completion of the group. If the commutator subgroup is one- or two-dimensional, we prove that our bound is in fact exact, and we conjecture that this holds in general.

Figures

Figures reproduced from arXiv: 2505.21090 by the authors.

Figure 1
Figure 1. Example illustrating the construction of the tuples of Lemma 6.11. There is one sub [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗

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