arxiv: 2605.05087 · v1 · submitted 2026-05-06 · 🧮 math.NT · math.AT· math.GR· math.GT

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The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings

Urshita Pal

Pith reviewed 2026-05-08 16:21 UTC · model grok-4.3

classification 🧮 math.NT math.ATmath.GRmath.GT

keywords cohomology of arithmetic groupsprincipal congruence subgroupsspecial linear groupsTits buildingsEuclidean ringsnumber theoryvirtual cohomological dimension

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The pith

For primes p in Euclidean number rings, the top cohomology of level-p congruence subgroups of SL_n always surjects onto the reduced homology of the quotient Tits building.

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

The paper studies the highest-degree cohomology of principal congruence subgroups inside special linear groups over Euclidean rings. It establishes that a natural comparison map from this top cohomology to the reduced homology of the Tits building quotient is always surjective when the level is a prime element. The work generalizes a question originally posed for the integers to other Euclidean rings such as the Gaussian integers. A reader would care because the result links the algebraic cohomology directly to a more geometric object, offering a route to explicit calculations of these groups. Under additional conditions on the prime, the map becomes an isomorphism, which would identify the cohomology group with that homology.

Core claim

We prove that for a prime p in a Euclidean number ring R with fraction field K, the natural map H^ν(Γ_n(p)) → H̃_{n-2}(T_n(K)/Γ_n(p)) is always surjective, and give a sufficient set of conditions on p ∈ R that guarantee when this map is an isomorphism. Here ν denotes the virtual cohomological dimension of the group, and T_n(K) is the Tits building associated to SL_n(K).

What carries the argument

The natural comparison map from the top cohomology of the congruence subgroup Γ_n(p) to the reduced homology of the quotient of the Tits building T_n(K) by the group action.

If this is right

The top cohomology group is computable from the homology of the building quotient whenever the extra conditions on p hold.
The result extends the Lee-Szczarba question from the ring of integers to all Euclidean number rings.
For primes satisfying the sufficient conditions the cohomology is completely determined by the building quotient.
Vanishing or non-vanishing statements for the cohomology follow directly from properties of the building quotient.

Where Pith is reading between the lines

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

The surjectivity might persist for certain non-prime levels if the generation arguments can be adapted.
Explicit calculations for small Euclidean rings could now be attempted by studying the building quotients instead of direct group cohomology.
The approach suggests a possible route to stability results for these cohomology groups as n grows.

Load-bearing premise

The ring R must be Euclidean so that SL_n(R) has good generation properties, and p must be a prime element.

What would settle it

A concrete computation for small n, a specific Euclidean ring R, and a prime p showing that the natural map fails to be surjective would disprove the surjectivity claim.

Figures

Figures reproduced from arXiv: 2605.05087 by Urshita Pal.

**Figure 1.** Figure 1: Decomposing triangles/suspends into sums of D-triangles, D-suspends, and multi-suspends. To view at source ↗

**Figure 2.** Figure 2: The disk bounded by the two simplices in the proof of Lemma view at source ↗

**Figure 3.** Figure 3: The sphere in the proof of Lemma 2.36 in the case n “ 3, along with the result of breaking it into n “ 3 spheres. we have a natural inclusion ι : BDAU n´1 pFq ãÑ BDAU n pFq induced by the inclusion x⃗v1, . . . , ⃗vn´1y ãÑ x⃗v1, . . . , ⃗vny. Then the map pι ˝ Hq ˚ Jr⃗vns,r⃗v1 ` ¨ ¨ ¨ ` ⃗vnsK : Dn´1 ˚ B∆1 – Dn Ñ BDAU n pFq gives us a homotopy from this map to the map g ˚ Jr⃗vns,r⃗v1 ` ¨ ¨ ¨ ` ⃗vnsK : B∆n´1 … view at source ↗

**Figure 4.** Figure 4: On the left is the sphere ρ ˝ f in the proof of Lemma 2.39 for n “ 3 with its three subdivided faces. On the right is the n “ 3 spheres it can be cut into (with the required subdivisions omitted to improve readability). For 1 ď i ď n ´ 1, these are the boundaries of additive simplices, and thus are trivially nullhomotopic in BDAU n pFq. So we need only deal with the i “ n case. Let u 1 P U be a unit so tha… view at source ↗

**Figure 5.** Figure 5: On the left is the sphere appearing in the case view at source ↗

**Figure 6.** Figure 6: The sphere in the proof of Lemma 2.40 in the case n “ 3, along with the result of breaking it into n “ 3 spheres. 2.3.5 Killing initial D-suspend maps In this section we prove Lemma 2.30, i.e. we show that the images of initial D-suspend maps in BDAU n pFq are nullhomotopic. The proof will require two lemmas. Lemma 2.40. Let F be a field, and U Ă F ˆ a subgroup of its units, such that: • The multiplicative… view at source ↗

**Figure 7.** Figure 7: The homotopy we are trying to achieve in Lemma view at source ↗

**Figure 8.** Figure 8: Elements of Zrωs with norm less than or equal to the norm of 4ω`1. Red points indicate multiples of 4ω ` 1 by units in t˘1, ˘ω, ˘ω 2 u, and blue points indicate elements that are ˘1, ˘ω or ˘ω 2 mod 4ω ` 1. is nullhomotopic. Let ρ 1 : BAOU n´1 pFq Ñ BDAU n´1 pFq be the retraction constructed in 2.35. Applying Lemma 2.41 to S 0 ˚ σ for each pn ´ 2q-simplex σ of S n´2 , we see that our map is homotopic to Jr⃗… view at source ↗

**Figure 9.** Figure 9: Elements of Zrωs with norm less than or equal to the norm of 4ω`3. Red points indicate multiples of 4ω ` 3 by units in t˘1, ˘ω, ˘ω 2 u, and blue points indicate elements that are ˘1, ˘ω or ˘ω 2 mod 4ω ` 3. Proof. The condition that tr⃗v1s,r⃗v2su is an edge in BDAU 2 pFq implies that detp⃗v1, ⃗v2q P U Since we are working with U-vectors, we can multiply ⃗v1 by a unit in U if necessary and assume that detp⃗v… view at source ↗

**Figure 10.** Figure 10: The process for using the vertex r ⃗ws to reduce the length of the loop γ, as described in the proof of Lemma 2.46. Proof. As before, we can act by a suitable element of SL2pFq and assume that ⃗vi “ ⃗ei for i “ 1, 2. The condition that tr⃗v0s,r⃗v1su is an edge of BDAU 2 pFq and that tr⃗v0s,r⃗v2su is not then implies that ⃗v0 “ c⃗e1 ` u⃗e2 for some c P F ˆzU and u P U. Suppose c “ u1 ` u2 for some u1, u2 P… view at source ↗

read the original abstract

For $R$ a Euclidean number ring, and let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $\Gamma_n(p)$ vanishes above a degree $\nu$ that is quadratic in $n$. Let $K$ be the fraction field of $R$, and $\mathcal{T}_n(K)$ the Tits building of $\text{SL}_n(K)$. For $R=\mathbb{Z}$, Lee--Szczarba asked when $\text{H}^\nu(\Gamma_n(p))$ is isomorphic to $\widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/\Gamma_n(p))$, which was answered by Miller--Patzt--Putman. We study a generalized version of Lee--Szczarba's question. We prove that for a prime $p$ in a Euclidean number ring $R$ with fraction field $K$, that a natural map $\text{H}^\nu(\Gamma_n(p)) \to \widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/\Gamma_n(p))$ is always surjective, and give a sufficent set of conditions on $p \in R$ that guarantee when this map is an isomorphism.

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 paper proves that for any prime element p in a Euclidean number ring R with fraction field K, the natural map H^ν(Γ_n(p)) → H̃_{n-2}(T_n(K)/Γ_n(p)) is surjective, where ν is the Borel-Serre vanishing degree and Γ_n(p) is the principal congruence subgroup of SL_n(R). It also supplies explicit sufficient conditions on p under which the map is an isomorphism. This generalizes the Lee-Szczarba question and the Miller-Patzt-Putman isomorphism result from the case R = Z.

Significance. If the surjectivity and conditional isomorphism statements hold, the work strengthens the connection between the top cohomology of congruence subgroups of SL_n over Euclidean rings and the homology of the associated Tits building quotients. It supplies a uniform surjectivity result that holds for every prime p (rather than only under restrictive arithmetic conditions) and identifies a concrete set of p for which the map becomes an isomorphism, thereby extending the range of rings and levels for which explicit computations of stable cohomology are feasible.

minor comments (3)

Abstract: the opening sentence contains a grammatical awkwardness ('For R a Euclidean number ring, and let Γ_n(p) be...'). A smoother phrasing such as 'Let R be a Euclidean number ring and let Γ_n(p) be...' would improve readability.
Abstract: 'sufficent' is a typographical error and should be corrected to 'sufficient'.
Abstract: the statement that the map 'is always surjective' is clear, but the precise definition of the natural map (e.g., via the building resolution or the Borel-Serre spectral sequence) is not indicated; a single sentence sketching its construction would help readers locate the argument in the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, their accurate summary of the main results, and their recommendation for minor revision. We are pleased that the work is viewed as strengthening the connection between top cohomology of congruence subgroups and Tits building homology, and as providing a uniform surjectivity result across all primes p.

Circularity Check

0 steps flagged

No circularity: derivation builds on external theorems without self-referential reduction

full rationale

The paper proves surjectivity of the natural map H^ν(Γ_n(p)) → H̃_{n-2}(T_n(K)/Γ_n(p)) for primes p in Euclidean R, with isomorphism under stated conditions on p. It invokes Borel-Serre vanishing (external) and the Miller-Patzt-Putman isomorphism over Z (external, different authors) as starting points, then constructs the generalized map via the Tits building resolution. No equation or claim reduces by definition to a fitted parameter, self-citation, or prior result by the same author; the hypotheses (Euclidean R, prime p) are standard and directly enable the generation properties used. The central claim has independent content beyond the cited inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the Euclidean property of R (to ensure generation and building properties) and standard facts about Tits buildings and group cohomology vanishing.

axioms (2)

domain assumption R is a Euclidean number ring with fraction field K
Invoked to define Γ_n(p) and the Tits building T_n(K); appears in the setup of the main theorem.
standard math Borel-Serre vanishing theorem for cohomology above degree ν
Used to identify the top degree ν; cited in the abstract.

pith-pipeline@v0.9.0 · 5538 in / 1341 out tokens · 24910 ms · 2026-05-08T16:21:29.601183+00:00 · methodology

discussion (0)

Reference graph

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