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Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures

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Pith's one-line read Steinberg homology vanishes in a range for every reductive group.

desk verdict Serious, substantial paper with Theorem B extending Steinberg vanishing to all reductive groups; Theorem D is new but rests on a deferred exactness proof and a thinly sketched case, so it needs revision, not desk rejection. read the letter →

arxiv 2509.01559 v1 pith:UWW5BKZZ submitted 2025-09-01 math.AT math.GRmath.RT

Jeremy Miller , Peter Patzt , Andrew Putman This is my paper · ORCID
classification math.ATmath.GRmath.RT MSC 20J0620G1011F7551E24
keywords SteinbergrepresentationreductivegroupshomologicalvanishingTitsbuildingrelativerootsystemsarithmeticdoubleSL_n(Z)cohomology
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

The paper's central claim is that for any connected reductive group G over a field k, the group homology of G(k) with coefficients in the Steinberg representation vanishes up to a degree depending only on the relative root system of G, and this holds over every commutative coefficient ring. Previous arguments covered the split classical groups; here the proof is reorganized around a resolution whose terms are built from the Steinberg representations of Levi subgroups, plus a spectral sequence whose differentials can be computed. The same construction is carried to the integers: the paper states a connectivity conjecture about the double Tits building of pairs of compatible flags of direct summands of Z^n, and shows this connectivity would yield the long-sought vanishing in the stable range for SL_n(Z). It proves the required connectivity for the first nontrivial case, showing the double building is n-connected for n at least 4, and thereby refines what is known in low degrees for GL_n(Z) and SL_n(Z) with Steinberg coefficients.

What carries the argument

The engine is a resolution of St(G) whose degree-i term is a sum, over standard Levi subgroups obtained by deleting i+1 simple roots, of induced Steinberg representations of those Levi subgroups. Feeding this resolution into group homology gives a spectral sequence whose E^1 page is built from smaller-rank groups; the induction is driven by the parabolic induction formula for Steinberg representations and by vanishing and surjectivity statements for the end Levi factors. For the integer theorems the load-bearing object is the double Tits building T^2(Z^n), the simplicial complex of pairs of compatible flags of direct summands; a prior theorem identifies its homology with that of the bar reso

What would settle it

Search the finite generating sets of Examples 31.3 and 31.4 for rank 4 or 5: if any element of X_1(V) is killed by the boundary to X_0(V) but is not in the image of the boundary from X_2(V), then Theorem 31.5 and hence Theorem D are false. Equivalently, a direct computation of H_1 of the bar resolution S_bullet(Z^4) that produced a nonzero class would settle the question in the negative.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that homological vanishing for Steinberg representations is a formal consequence of the structure of relative root systems, not of special features of split groups. Theorem B states that H_i(G(k); St(G;F)) = 0 for i at most b(Phi_k(G)), with b(A_n) = floor((n-1)/2), b(B_n) = b(C_n) = b(BC_n) = floor((n-2)/2), b(D_n) = floor((n-3)/2), b = 0 for exceptional systems, and for reducible systems a sum of the component bounds plus m-1. In the integral setting, Theorem D states that the double Tits building T^2(Z^n) is n-connected for n at least 4, and combined with a conditional spectral sequence it yields Corollary E: H_1 and H_2 with Steinberg coefficien

Load-bearing premise

For the integral half, the load-bearing premise is the exactness of the truncated three-step resolution X_2(V) -> X_1(V) -> X_0(V) -> St(V) -> 0; the paper verifies it by saying the generators and relations match a cited source and that enlarging X_2 does not affect the claim, rather than giving a full detailed proof, and if this exactness fails the connectivity of T^2(Z^n) would collapse.

Editorial extensions

If this is right

  • Theorem B gives a vanishing range for nonsplit reductive groups, so Steinberg-cohomology vanishing now applies to every connected reductive group over any field.
  • The proof replaces topology of partial flag complexes with a self-contained spectral sequence from a Levi-subgroup resolution, and the same machinery drives the integral results.
  • For exceptional relative root systems the theorem yields only H_0 vanishing, but the paper notes that a less uniform argument would raise the bounds for E_6, E_7, and E_8.
  • The n-connectivity of T^2(Z^n) for n at least 4 yields new integral vanishing: H_1 for SL_n(Z) and GL_n(Z) in ranges starting near n=4 or 5, and H_2 near n=6 or 8, depending on whether 2 and 3 are invertible in the coefficient ring.
  • If the connectivity assumption in Theorem C holds for larger b, the same mechanism proves the conjectured stable-range vanishing for SL_n(Z) and GL_n(Z) up to degree b for large n.

Reading between the lines

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

  • My inference: the relative-root-system formulation suggests analogous integral vanishings for other reductive Z-forms, such as symplectic or orthogonal groups, should follow from a connectivity conjecture for the appropriate integral double buildings; the paper explicitly says the approach can be generalized but does not write the details.
  • My inference: the 1/3 slope in the integral theorem appears forced by the base case SL_3(Z), where surjectivity is only verified after inverting 2 and 3; a sharper base case or a different resolution might make a 1/2 slope accessible, but the paper's own remark indicates its spectral sequence alone cannot achieve that.
  • My inference: the combinatorial restatement in Lemma 33.1 turns each summand of the connectivity proof into a finite statement about partitions of a set respecting specified subsets, so low-rank cases could be checked by computer search and might indicate whether higher cases of the b-integral resolution conjecture hold.
  • My inference: a natural testable extension is to replace Z by Z[1/N] or rings of integers in number fields; the bar-resolution formalism and the double-building definition carry over, and the first obstruction would be the analogue of the three-step partial resolution for those coefficient rings.
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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

3 major / 4 minor

Summary. The paper proves that for any connected reductive group G over a field k, the homology of G(k) with coefficients in the Steinberg representation vanishes in a range depending only on the relative root system Φ_k(G) (Theorem B). For the classical split groups this was known by Ash–Putman–Sam; the new proof is uniform and extends to all reductive groups, including nonsplit forms and the nonreduced system BC_n. The paper also states an integral refinement: assuming a high-connectivity conjecture for the double Tits building T^2(Z^n), it obtains vanishing for H_i(SL_n(Z); St) and H_i(GL_n(Z); St) in a range that improves with the connectivity assumption (Theorem C). The main new unconditional evidence is Theorem D, asserting that T^2(Z^n) is n-connected for n ≥ 4, which yields, via Theorem C, new low-degree vanishing for SL_n(Z) and GL_n(Z) (Corollary E). The proofs use a resolution of the Steinberg representation, a spectral sequence built from Levi subgroups, and, for the integral part, a double complex of partial resolutions.

Significance. If the gaps identified below are repaired, this is a substantial contribution. Theorem B unifies and extends prior vanishing results to all reductive groups over fields, with explicit bounds depending on the relative root system; the spectral sequence machinery and the explicit differential calculations in Parts 2–4 are detailed and appear sound. The integral part gives a clean conditional framework for the Church–Farb–Putman conjecture and proves a genuine new connectivity result for the double Tits building, leading to new low-degree vanishing for SL_n(Z) and GL_n(Z). The paper is careful to attribute prior work and states the precise dependency on external results (Reeder’s theorem, Solomon–Tits, and Miller–Patzt–Wilson’s identification of the bar resolution with the double Tits building). These strengths are real, but the load-bearing omissions discussed in the major comments must be addressed before the result can be considered fully verified.

major comments (3)
  1. [§31.3, Theorem 31.5; used in §32.2, Lemma 32.2] The exactness of X2(V)→X1(V)→X0(V)→St(V)→0 is asserted by comparison with [9], with the comment that enlarging X2 does not affect the claim because no assertion is made about ker(X2→X1). This is load-bearing: Lemma 32.2 requires H_i(X•(V))=0 for i=0,1 for every summand, and Theorem D′ (hence Theorem D and Corollary E) would fail if H_1(X•(V)) were nonzero. The argument is valid only if the enlarged X2(V) is a genuine chain complex with δ²=0 and its image lies in ker(X1→X0), and if the image of the [9] X2-term already covers that kernel. None of these is demonstrated; the text refers only to Examples 31.2–31.4. Please provide a direct verification for the new generators (especially the multi-line forms in Example 31.4) or quote and verify the precise statement from [9].
  2. [§25.4 (Theorem C′.1)] The proof of Theorem C′.1 is omitted with the note that it follows Part 2 closely after replacing the spectral sequence and the reducible-Levi results. This theorem supplies the 2-and-3-invertible cases of Corollary E, one of the paper's main concrete applications. The listed replacements are plausible, but the key differential lemmas (the analogues of Lemmas 12.1–12.3) are not stated or checked, and the induction with the cap min(b,·) needs verification. I recommend writing out the full proof or at least an appendix containing the integral analogues of the differential lemmas with precise statements.
  3. [§1.9, Theorem C] The statement of Theorem C gives the general-coefficient vanishing range as i ≤ min(b, ⌊(n−3)/3⌋). This is inconsistent with Theorem C′ (§25.2), which for GL_{n+1} gives i ≤ min(b, ⌊(n−1)/3⌋), i.e., for GL_n, i ≤ min(b, ⌊(n−2)/3⌋). Corollary E's thresholds n≥5 for i=1 and n≥8 for i=2 match the latter. The formula in Theorem C should be corrected to ⌊(n−2)/3⌋ (or the statement of Theorem C′ should be aligned if the intended bound is different).
minor comments (4)
  1. [Abstract] Typo: “The generalizes work” should be “This generalizes work”; “first and second of homology” should be “first and second homology groups.”
  2. [§1.9, Remark 1.18] The table and surrounding text are accurate but the sentence “In fact, with only a little more effort the proofs in [9, 15] work for F a commutative ring in which all primes p ≤ n are invertible” is slightly imprecise: the statement is correct for the specific claims used, but the reader must infer the exact scope. Consider making the coefficient condition explicit in the table.
  3. [§30.4, formula for ∂] The boundary formula for the bar complex is written with signs (−1)^{j−1}; this is fine, but the signs in the double-complex differential δ in §32.2 (with the shifts by r_1+...+r_{j−1}+i_1+...+i_{j−1}) deserve a small justification, especially because Lemma 32.2 relies on the signed differentials being compatible with the tensor-product differentials. Currently the verification is only hinted at.
  4. [Throughout] The paper is very long and some parts are repetitive (e.g., the three parallel treatments of types A, B/C/BC, and D). This is a presentation issue rather than a correctness issue.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main derivation is self-contained and the cited same-author results are used as independent external theorems; the terse proof of Theorem 31.5 is a verification risk, not a circular reduction.

full rationale

The paper's central vanishing theorem (Theorem B) is proved from a resolution of the Steinberg representation, the Solomon–Tits theorem, Reeder's induction theorem, standard reductive-group structure theory, and an induction on rank with explicit differential computations. None of these inputs is defined in terms of the vanishing conclusion, and no fitted parameter is later renamed as a prediction. The integral part is explicitly conditional: Theorem C assumes a connectivity hypothesis, and Theorem D proves a special case of that hypothesis. Theorem D does not assume the n-connectivity of T^2(Zn); instead it proves exactness of a three-step resolution and compares double complexes. The load-bearing inputs Theorem 23.3 (from Miller–Patzt–Wilson [22]) and Theorem 31.5 (from Brück–Miller–Patzt–Sroka–Wilson [9]) are external published results with overlapping authorship, but their statements do not include the target connectivity theorem and they are used as black boxes with independent proofs. The proof of Theorem 31.5 is notably terse: it says the enlarged X2 does not affect the exactness proved in [9], without displaying the verification that the added X2 generators lie in the kernel of the X1 differential. If that exactness failed, Lemma 32.2 and hence Theorem D would fail; this is a genuine correctness/verification risk in the manuscript, but it is not a circularity because the argument is an appeal to an external theorem, not an assumption of the theorem being proved. No uniqueness theorem is imported to forbid alternatives, and no known result is merely renamed. Accordingly, the score stays in the non-circular 0–2 range, with the small elevation reflecting the same-author citations and the terse delegation rather than any reduction-by-construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs depend on deep structural theorems about reductive groups and Steinberg modules, and on three external results with overlapping authorship. These are used as black boxes rather than reproved. No fitted parameters or proposed physical entities are introduced.

assumptions (5)
  • standard math Solomon-Tits theorem: the Tits building T(G) is homotopy equivalent to a wedge of (n-1)-spheres, so St(G;F) = H_{n-1}(T(G);F).
    Used throughout, e.g., in Section 1.2 and in the proof of Proposition 4.1 to build the resolution of St(G).
  • standard math Borel-Tits structure theory for reductive groups: existence and conjugacy of parabolic subgroups, Levi decomposition, relative root systems, BN-pairs, Bruhat decomposition, and root subgroups.
    Invoked in Sections 2.2, 2.7, 2.8, 2.13 and used to set up the Tits building and the resolution.
  • domain assumption Reeder map (Theorem 2.7): for a parabolic P with Levi factor L, St(G)|_{P(k)} is isomorphic as a P(k)-module to Ind_{L(k)}^{P(k)} St(L).
    This external theorem is the key structural input that turns the complex of parabolic subgroups into a resolution by induced representations; cited from [25].
  • domain assumption Miller-Patzt-Wilson Theorem 23.3: H_i(S_*(Z^{n+1})) is isomorphic to H_{i+n}(T^2(Z^{n+1})) for the bar resolution of the Steinberg module.
    This bridges the algebraic bar resolution to the topology of the double Tits building; taken from the authors' prior paper [22] and used in Corollary 23.4 and the proof of Theorem D.
  • domain assumption Bruck-Miller-Patzt-Sroka-Wilson resolution: the two-step presentation X_1(V) -> X_0(V) -> St(V) -> 0 is exact, and the variant Theorem 31.5 extends it to a three-step complex with X_2(V).
    This result is the foundation of Part 6; it is cited from [9] and verified in the paper only by referring to the matching generators and relations.

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Pith. "Pith review of Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures." pith.science (2026). https://pith.science/paper/UWW5BKZZ

@misc{pith2026250901559,
  author       = {Pith},
  title        = {Pith review of: Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWW5BKZZ}},
  note         = {Machine review of arXiv:2509.01559}
}
abstract

We prove that the homology groups of any connected reductive group over a field with coefficients in the Steinberg representation vanish in a range. The generalizes work of Ash-Putman-Sam on the classical split groups. We state a connectivity conjecture that would allow us to prove such a vanishing result for $SL_n(\mathbb{Z})$, as was conjectured by Church-Farb-Putman. We prove some special cases of this conjecture and use it to refine known results about the first and second of homology of $SL_n(\mathbb{Z})$ with Steinberg coefficients.

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