The Atiyah conjecture and \(L^2\)-cohomology computations
Let \(G\) be a discrete group and \(X\) a proper cocompact \(G\)-\(CW\)-complex. The \textit{strong Atiyah conjecture} states that \(\operatorname{lcm}G\cdot b^{(2)}_{k}(X;G)\) is an integer, where \(\operatorname{lcm}G\) denotes the least common multiple of orders of finite subgroups of \(G\) and \(b^{(2)}_{k}(X;G)\) denotes the \(k\)th-\(\ell^{2}\) Betti number of \(X\). This paper extends the validity of the strong Atiyah conjecture to some classes of groups and spaces including certain Coxeter groups and fundamental groups of special cube complexes which are quasi-isometric to a wedge of Euclidean half spaces. A nonpositively curved complex is \textit{sparse} if it is quasi-isometric to a wedge of finitely many Euclidean half-spaces. A cube complex \(X\), admits a \textit{cubical hierarchy} if \(X\) can be cut along embedded hyperplanes to a disjoint union of compact contractible cube complexes. The main result of this paper is the following: Theorem. Let \(G\) be the fundamental group of a sparse cube complex \(X\) that admits a cubical hierarchy and is hyperbolic relative to the collection of peripheral subgroups \(\mathcal{P}\). Suppose each peripheral subgroup is free abelian. Then, a finite index subgroup of \(G\) is a retract of a virtually cocompact special group \(G\). Hence, any group commensurable to \(G\) satisfies the strong Atiyah conjecture. As a consequence of this result, the authors prove vanishing results of \(\ell^{2}\)-homology for certain Coxeter groups.
scientific article; zbMATH DE number 6924057
scientific article