The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914535404732416 |
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| author | Pal, Urshita |
| author_facet | Pal, Urshita |
| contents | For $R$ a Euclidean number ring, and let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $Γ_n(p)$ vanishes above a degree $ν$ 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 $\text{H}^ν(Γ_n(p))$ is isomorphic to $\widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$, 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 $\text{H}^ν(Γ_n(p)) \to \widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$ is always surjective, and give a sufficent set of conditions on $p \in R$ that guarantee when this map is an isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05087 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings Pal, Urshita Number Theory Algebraic Topology Group Theory Geometric Topology For $R$ a Euclidean number ring, and let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $Γ_n(p)$ vanishes above a degree $ν$ 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 $\text{H}^ν(Γ_n(p))$ is isomorphic to $\widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$, 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 $\text{H}^ν(Γ_n(p)) \to \widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$ is always surjective, and give a sufficent set of conditions on $p \in R$ that guarantee when this map is an isomorphism. |
| title | The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings |
| topic | Number Theory Algebraic Topology Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2605.05087 |