Johnson, F. E. A. Fibrations of locally symmetric spaces and the failure of the Jordan- Hölder property. (English) Zbl 0631.53041 Proc. Am. Math. Soc. 98, 287-293 (1986). 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 Cited in 2 Documents MSC: 53C35 Differential geometry of symmetric spaces 22E40 Discrete subgroups of Lie groups 20E15 Chains and lattices of subgroups, subnormal subgroups Keywords:irreducible discrete subgroup; Jordan-Hölder property; locally symmetric space PDFBibTeX XMLCite \textit{F. E. A. Johnson}, Proc. Am. Math. Soc. 98, 287--293 (1986; Zbl 0631.53041) Full Text: DOI