  We study group actions on manifolds that admit hierarchies, which generalizes
the idea of Haken n-manifolds introduced by Foozwell and Rubinstein. We show
that these manifolds satisfy the Singer conjecture in dimensions $n \le 4$. Our
main application is to Coxeter groups whose Davis complexes are manifolds; we
show that the natural action of these groups on the Davis complex has a
hierarchy. Our second result is that the Singer conjecture is equivalent to the
cocompact action dimension conjecture, which is a statement about all groups,
not just fundamental groups of closed aspherical manifolds.
