The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^{\ast }(X)$ is the operator norm closure of the controlled operators on ${\mathrm{\ell }}^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^{\ast }(G,\mathcal{E}))$ and $K_1(C_u^{\ast }(G,\mathcal{E}))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\le G$, where $\mathcal{E}$ is the coarse structure generated by $H$. We use the Proietti–Yamashita spectral sequence to express the $K$-theory in terms of $H_{\ast }(H;{\mathrm{\ell }}^{\mathrm{\infty }}(G,\mathbb{Z}))$, which we then compute.