Universal Localizations, Atiyah Conjectures and Graphs of Groups

Abstract

Let G be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups satisfying the strong Atiyah conjecture over \(K \subseteq \mathbb{C}\) a field closed under complex conjugation. Assume that the orders of finite subgroups of G are bounded above. We show that G satisfies the strong Atiyah conjecture over K. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the ∗-regular closure of K[G] in \(\mathcal{U}(G)\), \(\mathcal{R}_{K[G]}\), is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding ∗-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over K are also closed under the graph of groups construction as long as the edge groups are finite. We also infer some consequences on the structure of the K₀ and K₁-groups of \(\mathcal{R}_{K[G]}\). The techniques developed enable us to prove that K[G] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that \(\mathcal{R}_{K[G]}\) is the universal localization of K[G] over the set of all matrices that become invertible in \(\mathcal{U}(G)\), provided that G belongs to a certain class of groups \(\mathcal{T}_{\mathcal{VLI}}\), which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.