Seminario Teoría de Grupos
THESIS PRE-DEFENSE -- L^2-Betti numbers, Strong Atiyah Conjecture and division ring embeddings for group algebras
Ponente: Pablo Sánchez Peralta (UAM)Fecha: jueves 05 de marzo de 2026 - 11:30Lugar: Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Resumen:
The study of \(L^2\)-Betti numbers of countable discrete groups has led to a rich homological theory with deep implications for the structural properties of groups. In this talk, we present recent, powerful tools to tackle longstanding problems regarding the vanishing and rationality of \(L^2\)-Betti numbers, as well as the structure and division ring embeddings of group algebras.
First, we introduce the novel notion of rk-Betti numbers, which serves as a unifying algebraic framework. Building on this concept, we explore new vanishing criteria for \(L^2\)-Betti numbers and present an algebraic proof of D. Gaboriau's well-known vanishing theorem.
Next, we address the Strong Atiyah Conjecture. We confirm that this conjecture is stable under taking free products—answering a longstanding open question—and we discuss how universal localizations explicitly describe the associated division rings.
Finally, we prove the existence of division ring embeddings for the group algebras of torsion-free virtually compact special groups and 3-manifold groups, showing that these families satisfy the Kaplansky Zero Divisor Conjecture.
