Measurable bounded cohomology of $t$-discrete measured groupoids via resolutions

Abstract

We define bounded cohomology of $t$-discrete measured groupoids with coefficients into measurable bundles of Banach spaces. Our approach via homological algebra extends the classic theory developed by Ivanov and by Monod. As a consequence, we show that the bounded cohomology of a $t$-discrete groupoid $\mathcal{G}$ can be computed using any amenable $\mathcal{G}$-space. In particular, we can compute bounded cohomology using strong boundaries.