Universal Localizations, Atiyah Conjectures and Graphs of Groups

• Published in: Geometric and functional analysis Vol. 35; no. 3; pp. 842 - 876
• Main Author: Sánchez-Peralta, Pablo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2025; Springer Nature B.V
• Subjects: Analysis; Conjugation; Fields (mathematics); Group theory; Localization; Mathematics; Mathematics and Statistics; Subgroups
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-025-00710-4

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 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 , , 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 . The techniques developed enable us to prove that K [ G ] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that is the universal localization of K [ G ] over the set of all matrices that become invertible in , provided that G belongs to a certain class of groups , which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.