On the cuspidal cohomology of arithmetic subgroups of SL(2n) and the first Betti number of arithmetic 3-manifolds
type: Item
instance of: scholarly article
title: On the cuspidal cohomology of arithmetic subgroups of SL(2n) and the first Betti number of arithmetic 3-manifolds
zbMATH Open document ID: 0648.22007
DOI: 10.1215/S0012-7094-87-05525-6
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: In the last decade much attention has been devoted to the cohomology of arithmetic subgroups of SL(2,k) when k is a number field with nice arithmetic properties. It was in particular shown that cuspidal cohomology exists if the arithmetic subgroup is deep enough. In this note the author proves the existence of cuspidal cohomology for deep arithmetic subgroups of SL(2,k) for any number field k. He then uses this result to show that a compact, hyperbolic 3-manifold \(Y=\Gamma \setminus SL(2,{\mathbb{C}})/SU(2)\) with arithmetic \(\Gamma\) has a finite covering \(\tilde Y\) with \(H^ 1(\tilde Y,{\mathbb{C}})\neq 0\). In the last paragraph he shows how to extend his construction of cuspidal cohomology to arithmetic subgroups of SL(2n,k), where k is again any number field.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 4057831
zbMATH Keywords: cuspidal cohomology
zbMATH Keywords: deep arithmetic subgroups
zbMATH Keywords: number field
zbMATH Keywords: finite covering
author: Laurent Clozel
reviewed by: Birgit Speh
MaRDI profile type: Publication
cites work: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120)
cites work: Q5566638
cites work: Q3927897
cites work: Q3049039
cites work: Q3885948
cites work: Regularization theorems in Lie algebra cohomology. Applications
cites work: Q4075116
cites work: Relative Lie algebra cohomology and unitary representations of complex Lie groups
cites work: Automorphic forms on GL (2)
cites work: On Euler Products and the Classification of Automorphic Representations I
cites work: On liftings and cusp cohomology of arithmetic groups
cites work: Unitary representations of Gl(n,R) with non-trivial (g,K)-cohomology
cites work: Q3235391
cites work: Solvable and Nonsolvable CM-Fields
full work available at URL: s0012-7094-87-05525-6
OpenAlex ID: W1572735334
description: scientific article; zbMATH DE number 4057831