A lower bound to the action dimension of a group (Q1880632)

The \textit{action dimension} of a discrete group is defined as the smallest dimension of a contractible manifold on which the group admits a properly discontinuous action (for example, the action dimension is \(m\) if the group acts properly discontinuously and cocompactly on a contractible \(m\)-manifold). In previous work [Invent. Math. 150, 219--235 (2002; Zbl 1041.57016)], \textit{M. Bestvina, M. Kapovich} and \textit{B. Kleiner} obtained a lower bound for the action dimension of a group by considering van Kampen's obstruction theory for embeddings of finite \(n\)-complexes into \(\mathbb R^{2n}\). They called complexes where this obstruction for an embedding into \(\mathbb R^m\) does not vanish \textit{\(m\)-obstructor complexes} (van Kampen's example for \(m=2n\) is the join of \(n+1\) sets of 3 points), introduced the notion of the \textit{obstructor dimension} of a discrete group and proved that this is a lower bound for the action dimension (as a typical application, when the group is hyperbolic or CAT(0) and its boundary contains an \(m\)-obstructor complex then the obstructor dimension is at least \(m+2\)). In the present paper the author refines this to the notion of a proper obstructor dimension; the main result states that this is again a lower bound for the action dimension. Using this extended notion and answering a question in the above cited paper he shows that a direct product of fundamental groups of a finite number of compact aspherical manifolds, with all boundary components aspherical and incompressible and some boundary component with fundamental group of index greater than 2 for each factor, the action dimension is the sum of the dimensions of the factors. Also, the proper obstructor dimension of the fundamental group of a closed aspherical \(m\)-manifold is equal to \(m\), its univeral cover being an \(m\)-proper obstructor.