Dynamical complexity and controlled operator \(K\)-theory (Q6599502)

Let a countable discrete group \(\Gamma\) acts on a compact Hausdorff space \(X\). The paper is based around a new property for actions named \textit{finite dynamical complexity}. For a set \(\mathcal C\) of open subgroupoids of \(\Gamma\ltimes X\), an open subgroupoid \(G\) of \(\Gamma\ltimes X\) is \textit{decomposable over} \(\mathcal C\) if for any \(r\geq 0\) there exists an open cover \(G^{(0)}=U_0\cup U_1\) of the unit space of \(G\) such that the subgroupoid of \(G\) generated by \(\{(gx,g,x)\in G:x\in U_i,|g|\leq r\}\) is in \(\mathcal C\) for \(i=0,1\), where \(|\cdot|\) denotes a fixed proper length function on \(\Gamma\). An open subgroupoid \(G\) of \(\Gamma\ltimes X\) has \textit{finite dynamical complexity} if it is contained in the smallest set \(\mathcal D\) of open subroupoids of \(\Gamma\ltimes X\) that contains all relatively compact open subgroupoids, and if \(G\) is decomposable over \(\mathcal D\) then \(G\in\mathcal D\).\N\NThis notion allows to compute the \(K\)-theory groups of \(C(X)\rtimes\Gamma\) by computing the \(K\)-theory groups of essentially finite pieces by commutative methods of algebraic topology and then by using generalized (controlled) Mayer--Vietoris arguments to reassemble this into \(K_*(C(X)\rtimes\Gamma)\). Note that \(C(X)\rtimes\Gamma\) is often simple, so the classical Mayer-Vietoris technique does not work.\N\NThe main part of the paper is the proof of the Baum-Connes conjecture for \(\Gamma\) with coefficients in \(C(X)\) under the assumption that the action of \(\Gamma\) on \(X\) has finite dynamical complexity and \(X\) is second countable. First, it is shown that this conjecture follows from triviality of the \(K\)-theory groups of a certain \textit{obstruction} \(C^*\)-algebra, and then the controlled Mayer-Vietoris technique is used to prove this triviality. Although this result is not completely new, the paper provides a direct and self-contained proof.\N\NThe paper is followed by two appendices. The first one relates finite dynamical complexity to similar notions, e.g. to topological amenability, and is followed by a short list of open problems. The second one identifies the model for the Baum--Connes assembly map used in the paper with one of the standard models using \(KK\)-theory.