Appears in collection : Orbit equivalence and topological and measurable dynamics / Equivalence orbitale, dynamique topologique et mesurée

A (discrete) standard Borel groupoid $\mathcal{G}$ is an internal groupoid in the category of standard Borel spaces and Borel maps with countable fibers. If, in addition, the set of units of $\mathcal{G}$ is equipped with a Borel measure $\mu$ preserved by the arrows of $\mathcal{G}$, we call the pair ( $\mathcal{G}, \mu$ ) a measurepreserving standard Borel groupoid (m.p. SBG for short).

In mimesis of the dialogue between algebraic topology and group theory, we introduce the notion of a measure-preserving standard Borel CW complex (abbreviated as m.p. SBCW complex). The fundamental groupoid of a m.p. SBCW complex is an m.p. SBG. Every m.p. SBG admits a classifying space, unique up to (an appropriate notion of) homotopy equivalence.

The $n$-dimensional truncated Euler characteristic of a m.p. standard Borel groupoid $\mathcal{G}$ is defined as

$$ \chi_n(\mathcal{G}):=\inf (-1)^n \chi\left(K^{(n)}\right) $$

where the infimum is taken over all classifying spaces $K$ of $\mathcal{G}$, and $\chi\left(K^{(n)}\right)$ denotes the Euler characteristic of the $n$-skeleton of $K$ (defined in a way appropriate for SBCW complexes). Finally, the $n$ dimensional cost of a countable group $\Gamma$ is defined as the infimum of the $n$-dimensional truncated Euler characteristics of the action groupoids associated to p.m.p. actions of $\Gamma$.

We explore connections between the $n$-dimensional cost, the $n$ dimensional rank gradient and $\ell^2$-Betti numbers of a countable group. This is joint work with Miklós Abért and Damien Gaboriau.