Bounded cohomological induction for transverse measured groupoids

TL;DR

This paper extends bounded cohomology theory to transverse measured groupoids, establishing an induction isomorphism that generalizes known results for lattices and provides explicit computations for complex groupoids.

Contribution

It introduces a new induction isomorphism for measurable bounded cohomology of groupoids, generalizing the Eckmann-Shapiro isomorphism and enabling explicit cohomology calculations.

Findings

Bounded cohomology of groupoids is isomorphic to that of the underlying group for unimodular cases.

If the group is amenable, the associated groupoid is boundedly acyclic.

Explicitly computed non-trivial bounded cohomology groups for certain higher rank Lie groups.

Abstract

We establish an induction isomorphism in the context of measurable bounded cohomology of discrete measured groupoid, which generalizes the Eckmann-Shapiro isomorphism in bounded cohomology of lattices due to Burger and Monod. In our wider setting, the role of lattices is taken by the class of transverse measured groupoids $(\mathcal{G}, \nu)$ associated with a cross-section $Y$ in a pmp dynamical system $(X, \mu)$ of a lcsc group $G$ such that the associated hitting time process of $Y$ is locally integrable. Typical examples are given by pattern groupoids of strong approximate lattices. Under the assumptions that $G$ is unimodular we show that the measurable bounded cohomology of $(\mathcal{G}, \nu)$ is isomorphic to the continuous bounded cohomology of $G$ with coefficients in $\text{L}^{\infty}(X, \mu)$. As a consequence, if $G$ is amenable, then $(\mathcal{G}, \nu)$ is boundedly acyclic, and in general the restriction map $\text{H}_{\text{cb}}^\bullet (G; \mathbb{R}) \to \text{H}_{\text{mb}}^\bullet ((\mathcal{G}, \nu);\underline{\mathbb{R}})$ is injective. Moreover, it follows from known results in continuous bounded cohomology that if $G$ is a semisimple higher rank Lie group of Hermitian (respectively complex classical) type, then the second (respectively third) measurable bounded cohomology of $(\mathcal{G}, \nu)$ is generated by the restriction of the bounded K\"ahler class (respectively bounded Borel class). These are the first explicit computations of non-trivial bounded cohomology groups of measured groupoids which are not isomorphic to an action groupoid.