Generalization to arbitrary compact metrizable spaces

Determine whether the dynamical comparison results established for Deaconu–Renault groupoids associated to minimal surjective local homeomorphisms on finite-dimensional compact metrizable spaces generalize to arbitrary compact metrizable spaces.

Background

The paper proves dynamical comparison for Deaconu–Renault groupoids associated to minimal surjective local homeomorphisms of compact metrizable spaces with finite covering dimension. Finite-dimensionality enters through the thin boundary property, which is used to control the boundaries of a large open subgroupoid arising from a partial action of a nonabelian free group. The authors note that earlier results for minimal amenable actions of nonamenable groups did not require dimensionality assumptions, motivating the question of whether their Deaconu–Renault groupoid results can likewise be extended beyond finite-dimensional unit spaces.