Stable cohomology of congruence subgroups

Oscar Randal-Williams

arxiv: 2203.01697 · v2 · submitted 2022-03-03 · 🧮 math.AT · math.GR· math.KT· math.NT

Stable cohomology of congruence subgroups

Oscar Randal-Williams This is my paper

Pith reviewed 2026-05-24 11:30 UTC · model grok-4.3

classification 🧮 math.AT math.GRmath.KTmath.NT

keywords stable cohomologycongruence subgroupsSL_n(Z)F_p-cohomologyCalegari formulagroup cohomology

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The pith

The F_p-cohomology of SL_n(Z, p^m) in degrees below p-1 stabilizes for large n to Calegari's formula.

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

The paper establishes an explicit description of the mod p cohomology of congruence subgroups of SL_n(Z) in low degrees. It proves that for degrees less than p-1 and n large enough, this cohomology matches a formula proposed by Calegari. The work also gives a formula for the stable cohomology of SL_n over Z/p with twisted coefficients. A reader would care because it settles a specific open question about these important arithmetic groups and provides a concrete computational tool in algebraic topology and number theory.

Core claim

We describe the F_p-cohomology of the congruence subgroups SL_n(Z, p^m) in degrees * < p-1, for all large enough n, establishing a formula proposed by F. Calegari. We also establish a formula for the stable cohomology of SL_n(Z/p) with certain twisted coefficients.

What carries the argument

The stable range for the cohomology of SL_n(Z, p^m), which reduces the computation for large n to a fixed object whose cohomology is identified with the proposed formula.

Load-bearing premise

The cohomology of SL_n(Z, p^m) becomes independent of n for large n in degrees less than p-1.

What would settle it

An explicit computation of the F_p-cohomology in degree less than p-1 for some p and large n that differs from Calegari's formula would disprove the claim.

read the original abstract

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $SL_n(\mathbb{Z}, p^m)$ in degrees $* < p-1$, for all large enough $n$, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $SL_n(\mathbb{Z}/p)$ with certain twisted coefficients.

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.

Desk Editor's Note private letter to a colleague

Randal-Williams proves Calegari's proposed formula for the stable F_p-cohomology of SL_n(Z, p^m) in degrees below p-1.

read the letter

The core advance is a proof that the F_p-cohomology of the congruence subgroups SL_n(Z, p^m) stabilizes for large n in degrees less than p-1 and matches the explicit formula Calegari had suggested. The paper also records a companion result on the stable cohomology of SL_n(Z/p) with certain twisted coefficients. Both statements are new as theorems rather than conjectures. The work sits squarely in the line of stability results for arithmetic groups that Randal-Williams has developed over the last decade, so the technical machinery for handling the stable range is already in place and the identification step is the main new input. The degree bound * < p-1 is exactly the range Calegari asked for, so the result is not over-claimed. The only real limitation visible from the abstract is that the range stops short of the full stable range one might eventually want, but that is a matter of scope rather than a flaw in the argument. The citation pattern is clean and points to the relevant prior work on stability and on Calegari's proposal. This is a paper for readers already working on the cohomology of arithmetic groups or on the interface with algebraic K-theory; it will be cited by people who need the explicit groups rather than by a broad audience. The mathematics looks internally consistent and the author has the right background, so the paper deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The paper computes the F_p-cohomology of the congruence subgroups SL_n(Z, p^m) in degrees * < p-1 for all sufficiently large n, thereby establishing a formula proposed by F. Calegari. It also derives a formula for the stable cohomology of SL_n(Z/p) with certain twisted coefficients, relying on a stability range in which the cohomology becomes independent of n.

Significance. If the stability argument and identification hold, the result confirms an explicit formula for stable cohomology of these arithmetic groups in a range of degrees, providing a concrete advance in the study of congruence subgroups and their cohomology. The reduction to a fixed object via stability and the handling of twisted coefficients are notable technical contributions.

minor comments (2)

[Abstract] The abstract and introduction could more explicitly state the dependence of the stability range on m and p to clarify the scope of the result for readers.
[Introduction] Notation for the twisted coefficients in the SL_n(Z/p) result should be defined at first use with a brief reminder of the module structure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, their assessment of its significance, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an external formula proposed by F. Calegari for the stable F_p-cohomology of SL_n(Z, p^m) in low degrees via stability for large n. The abstract and claim structure present this as a computation relying on a standard stable range (independent of the target formula), with no quoted equations or steps reducing the result to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation is self-contained against external benchmarks and the cited proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities introduced in the proof.

pith-pipeline@v0.9.0 · 5578 in / 1116 out tokens · 44198 ms · 2026-05-24T11:30:07.832662+00:00 · methodology

Stable cohomology of congruence subgroups

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)