Fibrations of locally symmetric spaces and the failure of the Jordan-Hölder property

Abstract

Abstract. We construct examples of locally symmetric spaces which fibre, with both base space and fibre also locally symmetric, in two distinct ways so that neither fibre nor base of one is homeomorphic to either fibre or base of the other. The simplest example is a complex surface which fibres over a complex curve in two essentially distinct ways.In consequence, the Jordan-Holder property fails for large classes of irreducible discrete subgroups of Lie groups. 0. Introduction. Let ^ be a class of groups. Recall that by a poly-fé' filtration on a group T we mean a sequence (rr)0<r<" of subgroups of T with (1) (iv) s/G. The class of torsion free arithmetic subgroups in a linear algebraic group G, defined and almost simple over Q, and such that (GR)0 has no compact simple factors.Observe that SURFACE is just the class ^Pgl2(R) and that FREE is just the classGiven a poly-^ filtration (rr)0<r<", we call « the height of the filtration. In fact, in each case, we show that the Jordan-Holder property fails for groups of height 2. The results can be viewed geometrically, as providing examples of locally symmetric spaces which decompose as fibre bundles in which both base and fibre are locally