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 that satisfy 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 $\ast$-regular closure of $K[G]$ in $\mathcal{U}(G)$, $\mathcal{R}_{\small K[G]}$, is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding $\ast$-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 provided that the edge groups are finite. We also infer some consequences on the structure of the $K_0$ and $K_1$-groups of $\mathcal{R}_{\small K[G]}$. The techniques developed allow us to prove that $K[G]$ fulfills the strong, algebraic and center-valued Atiyah conjectures and that $\mathcal{R}_{\small K[G]}$ is the universal localization of $K[G]$ over the set of all matrices that become invertible in $\mathcal{U}(G)$ if $G$ lies in a certain class of groups $\mathcal{T}_{\small \mathcal{VLI}}$, which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.