ON THE TOP-DIMENSIONAL COHOMOLOGY OF ARITHMETIC CHEVALLEY GROUPS BENJAMIN BRÜCK YURI SANTOS REGO AND ROBIN J. SROKA

2025-05-02 1 0 598.26KB 8 页 10玖币

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ON THE TOP-DIMENSIONAL COHOMOLOGY OF

ARITHMETIC CHEVALLEY GROUPS

BENJAMIN BRÜCK, YURI SANTOS REGO, AND ROBIN J. SROKA

Abstract. Let Kbe a number ﬁeld with ring of integers Oand let Gbe

a Chevalley group scheme not of type E8,F4or G2. We use the theory

of Tits buildings and a result of Tóth on Steinberg modules to prove that

Hvcd(G(O); Q)=0if Ois Euclidean.

1. Introduction

In this article, we obtain the following result about the cohomology of arithmetic

groups:

Theorem 1.1. Let Kbe a number ﬁeld, Othe ring of integers in Kand Ga

Chevalley–Demazure group scheme of type An,Bn,Cn,Dn,E6or E7. If Ois Eu-

clidean, then the rational cohomology of G(O)vanishes in its virtual cohomological

dimension vcd = vcd(G(O)),

Hvcd(G(O); Q)=0.

As SLnand Sp2nare the simply-connected Chevalley–Demazure schemes of types

An−1and Cn, respectively, Theorem 1.1 is a common generalisation of results of

Lee–Szczarba [LS76] (for SLn(O)) and Brück–Patzt–Sroka [Sro21, Chapter 5] (for

Sp2n(Z), building on work of Gunnells [Gun00]).

There are two main ingredients in the proof of Theorem 1.1. The ﬁrst is work

of Borel–Serre [BS73] who proved that the groups in question are virtual Bieri–

Eckmann duality groups. This allows one to study their high-dimensional rational

cohomology by analysing their low-dimensional homology with coeﬃcients in the

so called Steinberg module. The second ingredient is a result of Tóth [Tót05]

that gives a generating set of this module. He shows that in the cases covered in

Theorem 1.1, the Steinberg module is cyclic as a G(O)-module. This generalises

results by Ash–Rudolph [AR79] and Gunnells [Gun00] in the cases of SLnand Sp2n,

respectively.

The Steinberg module can be described as the top-dimensional homology group

of an associated Tits building. Previous vanishing results in the settings of SLn

and Sp2nused explicit descriptions of the buildings that were speciﬁc for the cor-

responding types; see [CFP19, Sections 1.1 and 4] for type Aand [Sro21, Chapter 5

and Deﬁnition 60] for type C. We prove a building-theoretic generalisation of the

key step in [CFP19, Section 4] for all types; see Proposition 3.3. This enables us

to show that cohomology vanishing in the virtual cohomological dimension always

BB and RJS were partially supported by the Deutsche Forschungsgemeinschaft (DFG, German

Research Foundation) – Project-ID 427320536 – SFB 1442, as well as by Germany’s Excellence

Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics–Geometry–Structure. YSR

was partially supported by the German Research Foundation (DFG) through the Priority Program

2026 ‘Geometry at inﬁnity’, Project 62, while working on this project at the OvGU Magdeburg.

RJS was partially supported by the European Research Council (ERC grant agreement No.772960)

and the Danish National Research Foundation (DNRF92, DNRF151) as a PhD Fellow at the Uni-

versity of Copenhagen, and by NSERC Discovery Grant A4000 in connection with a Postdoctoral

Fellowship at McMaster University.

arXiv:2210.12784v2 [math.AT] 13 May 2024

2 BENJAMIN BRÜCK, YURI SANTOS REGO, AND ROBIN J. SROKA

follows if one can show that the Steinberg module is generated by “integral apart-

ment classes”; see Theorem 3.4. The generation by integral apartments classes, in

turn, is the content of Tóth’s result.

There are two assumptions in Theorem 1.1. The ﬁrst is that Gbe not of type E8,

F4or G2. This is due to the same hypothesis in Tóth’s work and comes from the fact

that, in these cases, there is no maximal parabolic subgroup whose unipotent radical

is abelian [Tót05, Section 5]. This makes certain computations harder in these cases

[Tót05, second paragraph after Theorem 2]. The second assumption is that Obe

Euclidean, which is also a restriction in Tóth’s work. However, Euclideanity seems

to be a natural assumption for a general statement in the style of Theorem 1.1. This

is among other things indicated by work of Miller–Patzt–Wilson–Yasaki [MPWY20]

who obtain non-vanishing results for G= SLnand certain non-Euclidean PIDs O.

The condition that Oshould at least be a PID is necessary in a strong sense, at

least for G= SLn[CFP19, Theorem D] and G= Sp2n[BH23, Theorem 1.1].

In type A, for the group SLn(O), even stronger vanishing results are already

known: Church–Putman [CP17] showed that the rational cohomology of this group

vanishes also one degree below its virtual cohomological dimension if O=Z, and

Kupers–Miller–Patzt–Wilson [KMPW22] proved the same result for Othe Gaussian

or Eisenstein integers. Brück–Miller–Patzt–Sroka–Wilson [BMP+22] extended this

to vanishing of the rational cohomology two degrees below the virtual cohomological

dimension for O=Z. These results conﬁrm parts of a conjecture by Church–Farb–

Putman [CFP14] who asked whether it was generally true that

(1.1) Hvcd(SLn(Z))−i(SLn(Z); Q)=0if i<n−1 = rk(SLn).

In light of Theorem 1.1, one is tempted to ask whether vanishing behaviour

similar to Eq. (1.1) might also occur for other arithmetic Chevalley groups.

Question 1.2. Let Kbe a number ﬁeld, Othe ring of integers in Kand Ga

Chevalley–Demazure group scheme. If Ois Euclidean, is it true that

Hvcd(G(O))−i(G(O); Q) = 0 for all i < rk(G)?

Currently, evidence for such a vanishing pattern is given by Theorem 1.1, the

above mentioned results in type Aand work of Brück–Patzt–Sroka [BPS23] in type

Cthat shows that Hvcd(Sp2n(Z))−1(Sp2n(Z); Q)is trivial for n≥2.

Acknowledgements. We are indebted to Petra Schwer for helpful discussions, and to

Paul Gunnells for helpful comments and pointing us to [Tót05]. We thank Peter Patzt,

Jeremy Miller and Jennifer Wilson for comments on earlier versions of this article and

Dan Yasaki for comments on computations in low dimensions. RJS would like to thank

his PhD advisor Nathalie Wahl for enlightening conversations and helpful feedback about

[Sro21, Chapter 5]. We thank the anonymous referee for their comments and suggestions.

2. Background

2.1. Coxeter groups and Coxeter complexes. Given a ﬁnite set S, consider a

symmetric matrix M= (ms,t)s,t∈Swhose diagonal entries equal one and all other

entries are ∞or integers greater than one. A group Wwith presentation

W=⟨s∈S|(st)ms,t = 1 for all s, t with ms,t <∞⟩

is called a Coxeter group. The pair (W, S)is the corresponding Coxeter system, and

Sis the Coxeter generating set. The rank of the system (W, S)is the cardinality

|S|of the given generating set. We write ℓ(w)for the word length of w∈W

with respect to the generating set S. The system (W, S)is called spherical if the

underlying Coxeter group Wis ﬁnite. The reader is referred to standard textbooks,

such as [Hum72,GP00,AB08], for further background on Coxeter groups.

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ONTHETOP-DIMENSIONALCOHOMOLOGYOFARITHMETICCHEVALLEYGROUPSBENJAMINBRÜCK,YURISANTOSREGO,ANDROBINJ.SROKAAbstract.LetKbeanumberfieldwithringofintegersOandletGbeaChevalleygroupschemenotoftypeE8,F4orG2.WeusethetheoryofTitsbuildingsandaresultofTóthonSteinbergmodulestoprovethatHvcd(G(O);Q)=0ifOisEuclidean.1...

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