On topological filtrations of groups (Q2400109)

According to \textit{S. G. Brick} and \textit{M. L. Mihalik} [Math. Z. 220, No. 2, 207--217 (1995; Zbl 0843.57003)], a cell-complex \(X\) is said to be quasi-simple filtered (QSF) if its universal cover can be approximated by simply-connected finite complexes. More precisely, \(X\) is QSF if for every finite subcomplex \(A\) of the universal cover \(\tilde X\) of \(X\) there is a cellular map \(f\) from a simply-connected finite complex to \(\tilde X\), which is a homeomorphism on \(f^{-1}(A)\). Brick and Mihalik [loc. cit.] showed that this property only depends on the fundamental group. Therefore one may say that a finitely presented group \(G\) is QSF if any finite complex \(P\) which has \(G\) as fundamental group has a QSF universal covering. The question investigated in the paper under review is whether there are finitely-presented groups which are not QSF. It is shown (Corollary~2.6) that locally solvable groups are QSF, provided they have large nilpotent quotients. In Theorem~2.8, more conditions are listed that force a group all of whose subgroups are finitely presented to be QSF.