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Fibrations of locally symmetric spaces and the failure of the Jordan- Hölder property. (English) Zbl 0631.53041

A class C of groups is said to possess the Jordan-Hölder property if, for any group \(\Gamma\) and for any sequence \(\{1\}=\Gamma_ 0\subset \Gamma_ 1\subset...\subset \Gamma_ n=\Gamma\) of its subgroups such that \(\Gamma_ r \triangleleft \Gamma_{r+1}\) and \(\Gamma_{r+1}/\Gamma_ r\in C\) for each r, the number n and the set of the quotients \(\Gamma_{r+1}/\Gamma_ r\) do not depend on the sequence \((\Gamma_ r)\). The goal of the paper is to prove that the following two classes of groups do not possess the Jordan-Hölder property: the class of torsion free irreducible cocompact lattices in a noncompact linear semisimple Lie group with finitely many connected components; the class of torsion free arithmetic subgroups in a linear algebraic group G, defined and almost simple over \({\mathbb{Q}}\), and such that \((G_{{\mathbb{R}}})^ 0\) has no compact simple factors. The proof is geometrical and reduces to constructing, for some locally symmetric space, two fiberings with distinct locally symmetric fibres and bases.
Reviewer: A.L.Onishchik

MSC:

53C35 Differential geometry of symmetric spaces
22E40 Discrete subgroups of Lie groups
20E15 Chains and lattices of subgroups, subnormal subgroups
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