The action dimension of a discrete group Γ is the smallest dimension of a contractible manifold which admits a proper action of Γ. Associated to any flag complex L there is a right-angled Artin group, A L . We compute the action dimension of A L for many L. Our calculations come close to confirming the conjecture that if an ℓ 2 -Betti number of A L in degree l is nonzero, then the action dimension of A L is ≥ 2l.

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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 ≤ 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.

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show these conditions hold for cocompact special groups so the Strong Atiyah conjecture holds for virtually cocompact special groups.

The action dimension of a discrete group G is the minimum dimension of a contractible manifold that admits a proper G‐action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including (1) Artin groups, (2) graph products of groups and (3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.