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

Filippo Sarti, Alessio Savini

TL;DR

This work extends bounded cohomology to $t$-discrete measured groupoids with coefficients in dual measurable bundles of Banach spaces, situating the theory in a homological-algebra framework inspired by Ivanov and Monod. By developing measurable bundles, disintegration, and strong resolutions by relatively injective bundles, the authors show that bounded cohomology can be computed using any amenable $ ext{G}$-space, including boundaries, and they establish a vanishing result in the amenable case. A central achievement is the construction of a robust cohomological theory for groupoids that mirrors the group case while accommodating fibred coefficients and measurability, enabling applications to rigidity phenomena and amenability criteria. The results provide a toolkit for computing $ ext{H}_{ ext{mb}}^{k}( ext{G}, ext{E})$ via amenable spaces and yield new proofs of known vanishing theorems through this dual-bundle, fibred-structure approach.

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.

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

TL;DR

This work extends bounded cohomology to -discrete measured groupoids with coefficients in dual measurable bundles of Banach spaces, situating the theory in a homological-algebra framework inspired by Ivanov and Monod. By developing measurable bundles, disintegration, and strong resolutions by relatively injective bundles, the authors show that bounded cohomology can be computed using any amenable -space, including boundaries, and they establish a vanishing result in the amenable case. A central achievement is the construction of a robust cohomological theory for groupoids that mirrors the group case while accommodating fibred coefficients and measurability, enabling applications to rigidity phenomena and amenability criteria. The results provide a toolkit for computing via amenable spaces and yield new proofs of known vanishing theorems through this dual-bundle, fibred-structure approach.

Abstract

We define bounded cohomology of -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 -discrete groupoid can be computed using any amenable -space. In particular, we can compute bounded cohomology using strong boundaries.

Paper Structure

This paper contains 11 sections, 20 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Acknowledgements.
  4. Preliminaries
  5. Measured groupoids
  6. Amenability
  7. Measurable Bundles of Banach spaces
  8. Definition and examples
  9. The disintegration isomorphism
  10. Homological algebra for measurable bundles
  11. The bounded cohomology of measured groupoids

Key Result

Theorem 2

Let ${\mathcal{G}}$ be a $t$-discrete measured groupoid, $(S,\tau)$ an amenable $\mathcal{G}$-space and $\mathcal{E}$ a measurable ${\mathcal{G}}$-bundle that is the dual of a separable measurable ${\mathcal{G}}$-bundle. Then we have a natural isomorphism for every $k\geq 0$.

Theorems & Definitions (66)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • ...and 56 more