On the top-dimensional cohomology groups of
congruence subgroups of SL(n, ℤ)

Miller, Jeremy; Patzt, Peter; Putman, Andrew

doi:10.2140/gt.2021.25.999

Cited by 9 publications

(7 citation statements)

References 21 publications

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“…The goal of this subsection is to prove the following, which generalises [24,Lemma 2.58] Remark 3.13. When the set of generators in Theorem 3.12 is empty, we mean that the group H𝑛−1 (𝐵 𝑚 𝑛 ) ≅ 0.…”

Section: Generating Set For the Top Homologymentioning

confidence: 88%

Improved homological stability for certain general linear groups

Kupers

Miller

Patzt

2022

Proceedings of London Math Soc

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Section: Generating Set For the Top Homologymentioning

confidence: 88%

Improved homological stability for certain general linear groups

Kupers

Miller

Patzt

2022

Proceedings of London Math Soc

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“…is (n − 4)-connected by Lemma 3.15. As in the proof of [MPP,Lemma 2.4] and the first step of the proof of Theorem 3.16, we can homotope the map φ to avoid the simplex {v 0 , v 1 , v 2 } without introducing new additive simplices to its image. Iterating this procedure, and the analogous procedure for externally additive simplices, produces a map to B m n (O).…”

Section: Then the Linkmentioning

confidence: 99%

“…We say an ideal J has the Lee-Szczarba property (see [LS76b,Page 28]) if the natural map Often, H n−2 (T n (K)/Γ n (J)) is computable (see [MPP,Table 1]) so an answer to this question could yield concrete calculations.…”

Section: Open Questionsmentioning

confidence: 99%

On the generalized Bykovskii presentation of Steinberg modules

Kupers¹,

Miller²,

Patzt³

et al. 2020

Preprint

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We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskiȋ gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

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“…In a different direction, in the case of a congruence subgroup Γ(p) ⊂ SL n /Q of prime level p, Miller et al [27] prove that the Γ(p)-invariant map Hn−2 (T SLn , Z) −→ Hn−2 (T SLn /Γ(p), Z), induced by the quotient map T SLn −→ T SLn /Γ(p), is surjective. Therefore, the size of the right hand side provides a lower bound for the space (st SLn ) Γ(p) , thus, for the dimension of H cd(Γ(p)) (Γ(p), Q).…”

Section: Comparison With Prior Work and Extensionsmentioning

confidence: 99%

Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$

Schwermer

2020

Preprint

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The cohomology H * (Γ, E) of a torsion-free arithmetic subgroup Γ of the special linear Q-group G = SLn may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology The latter space is decomposed according to the classes {P} of associate proper parabolic Q-subgroups of G. Each summand H * {P} (Γ, E) is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in {P}.The cohomology H * (Γ, E) vanishes above the degree given by the cohomological dimension cd(Γ) = n(n−1) 2. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {P} for which the corresponding summand H cd(Γ) {P} (Γ, E) vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H cd(Γ) {Q} (Γ, C). Finally, in the case of a principal congruence subgroup Γ(q), q = p ν > 5, p ≥ 3 a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes {Q}.

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On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)

Cited by 9 publications

References 21 publications

Improved homological stability for certain general linear groups

Improved homological stability for certain general linear groups

On the generalized Bykovskii presentation of Steinberg modules

Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$

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