We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken –manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions . 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.
"The $L^2$–(co)homology of groups with hierarchies." Algebr. Geom. Topol. 16 (5) 2549 - 2569, 2016. https://doi.org/10.2140/agt.2016.16.2549