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 K 0 and K 1 -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.