HK property and K0 splitting for Baumslag–Solitar groupoids G(m,n)

Determine, for integers m and n with gcd(m,n)=1, whether the exact sequence 0 → Z/(n−1)Z → K0(C*(G(m,n))) → Z/(m−1)Z → 0 splits, where G(m,n) is the ample groupoid associated to the self-similar action of the solvable Baumslag–Solitar group BS(1,m) on the alphabet Z/nZ. Equivalently, ascertain for which pairs (m,n) the HK property holds for G(m,n), i.e., K0(C*(G(m,n))) ≅ H0(G(m,n)) ⊕ H2(G(m,n)).

Background

In Section 8.2 the paper studies the groupoid G(m,n) associated to the self-similar action of the solvable Baumslag–Solitar group BS(1,m) on Z/nZ (with n relatively prime to m). The authors compute the homology groups: H0(G(m,n)) ≅ Z/(n−1)Z, H1(G(m,n)) ≅ Z/(m−1)Z ⊕ Z/(n−1)Z, H2(G(m,n)) ≅ Z/(m−1)Z, and Hq(G(m,n))=0 for q≥3.

They note that, using Pooya–Valette’s spectral sequence, there is an exact sequence 0 → Z/(n−1)Z → K0(C*(G(m,n))) → Z/(m−1)Z → 0. Whether this extension splits is tied to whether the HK property holds for the groupoid: splitting would give K0(C*(G(m,n))) ≅ H0(G(m,n)) ⊕ H2(G(m,n)).