Purely infinite locally compact Hausdorff \'{e}tale groupoids and their $C^*$-algebras

In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff \'{e}tale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff \'{e}tale minimal topological principal groupoids. This generalizes the dichotomy obtained by B\"{o}nicke-Li and Rainone-Sims. We also study the relation among our paradoxical comparison, $n$-filling property and locally contracting property appeared in the literature for locally compact Hausdorff \'{e}tale groupoids.