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.

References

It remains an interesting question whether the results presented in this work generalize to arbitrary compact metrizable spaces.

— Dynamical comparison for local homeomorphisms  (2608.13000 - Geffen et al., 13 Aug 2026) in Introduction

It is however open up to what extent groupoid comparison is automatic (at least in the amenable case).

— A groupoid model for Rørdam's finite-infinite $C^*$-algebra  (2610.01543 - Keuppens et al., 1 Oct 2026) in Section 1, Introduction, paragraph discussing strict and dynamical comparison

Problem 6.2.2. Define a notion of dynamical strict comparison which is meaningful to non-minimal dynamical systems. Can one obtain generalizations of the results at the end of Section 4.3 for this notion?

— Classifiability of crossed products  (2610.06586 - Gardella, 5 Oct 2026) in Problem 6.2.2, Section 6.2, p. 48

Question 6.2.3. Is there a minimal, amenable dynamical system which does not have dynamical comparison?

— Classifiability of crossed products  (2610.06586 - Gardella, 5 Oct 2026) in Question 6.2.3, Section 6.2, p. 48