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On the cuspidal cohomology of arithmetic subgroups of SL(2n) and the first Betti number of arithmetic 3-manifolds. (English) Zbl 0648.22007
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.
Reviewer: B.Speh

##### MSC:
 22E40 Discrete subgroups of Lie groups 20G10 Cohomology theory for linear algebraic groups 20G30 Linear algebraic groups over global fields and their integers 57T10 Homology and cohomology of Lie groups
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