I will discuss a few dimension-type invariants for discrete group actions on locally compact Hausdorff spaces, which are not assumed to be free, the most significant of which is something we call the long and thin cover dimension, motivated by work of Bartels, Luck and Reich. Those are used to show that the associated crossed product has finite nuclear dimension. The result applies to arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces, but also covers other classes of actions, such as hyperbolic groups acting on boundaries of hyperbolic complexes (e.g. the Gromov boundary), as well as certain allosteric actions of wreath products of finite abelian groups by Z^d.

This generalizes and puts in a more conceptual framework previous work of ours on non-free actions of the integers.

This is joint work with Jianchao Wu.

The talk will be in two parts. The first part will be a survey. The second part is intended only for the subset of the audience who is interested in going into somewhat more technical details of some selected section of this long preprint (which is over 100 pages).