Bounded cohomological induction for transverse measured groupoids
Tobias Hartnick, Filippo Sarti
TL;DR
This work extends Burger–Monod–Eckmann–Shapiro induction from lattices to transverse measured groupoids by introducing integrable transverse G-systems and proving an isometric isomorphism H_mb^•(( G, u); underline{R}) ≅ H_cb^•(G; L^ fty(X, μ)). The construction leverages measure equivalence between transverse groupoids and ambient groups, transfer spaces, and amenable resolutions to transport bounded cohomology classes from G to the groupoid and vice versa. Consequently, amenable G yields bounded acyclicity of the transverse groupoid, and in substantial cases (Hermitian or complex classical) low-degree classes correspond to the restricted bounded K"ahler or Borel classes, giving explicit nontrivial computations for measured groupoids not arising from action groupoids. The results connect to measured subsets and pattern groupoids of strong approximate lattices, enabling new pattern invariants and discretization results, and provide concrete explicit implementations via harmonic cocycles and Poisson boundaries. Overall, the paper broadens bounded cohomology methods to a rich class of measured groupoids, with rigidity and discretization implications across dynamics, aperiodic order, and higher-rank phenomena.
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 $(\mathcal{G}, ν)$ associated with a cross-section $Y$ in a pmp dynamical system $(X, μ)$ 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 $(\mathcal{G}, ν)$ is isomorphic to the continuous bounded cohomology of $G$ with coefficients in $\text{L}^{\infty}(X, μ)$. As a consequence, if $G$ is amenable, then $(\mathcal{G}, ν)$ is boundedly acyclic, and in general the restriction map $\text{H}_{\text{cb}}^\bullet (G; \mathbb{R}) \to \text{H}_{\text{mb}}^\bullet ((\mathcal{G}, ν);\underline{\mathbb{R}})$ 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 $(\mathcal{G}, ν)$ is generated by the restriction of the bounded Kähler 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.