Characterization of the T0 orbit space of the uniform Roe groupoid

Characterize when the orbit space of the uniform Roe groupoid of a metric space is T0.

Background

The paper studies type I and AF ideals in uniform Roe algebras through coarse geometry. For a subset A of a countable exact group Γ, the associated compact ideal I_A is defined using the invariant open subset generated by the clopen set corresponding to A. The authors relate type I behavior of these ideals to properties such as sparseness and asymptotic dimension zero.

The orbit-space question is connected to the type I problem because the paper notes that asymptotic dimension at least one forces the orbit space of the uniform Roe groupoid not to be T0, while the largest type I ideal is shown to be AF. A general characterization of when the orbit space is T0 remains unresolved in the cited literature.

References

The related problem of characterizing when the orbit space of the uniform Roe groupoid is $T_0$ is left open in Problem 2.26.

— On AF- and type I-ideals in certain crossed product C$^\ast$-algebras  (2609.11297 - Ravnanger, 10 Sep 2026) in Section “Uniform Roe algebras”, paragraph beginning “Following up on the discussion in the previous section regarding the largest type I-ideal”