Purely infinite $C^{\ast }$-algebras associated to \'etale groupoids

Let $G$ be a Hausdorff, 'etale groupoid that is minimal and topologically principal. We show that $C_r^{\ast }(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^{\ast }(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C_r^{\ast }(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C_r^{\ast }(G)$. We then show how this result applies to $k$-graph $C^{\ast }$-algebras. Finally, we investigate strongly purely infinite groupoid $C^{\ast }$-algebras.