Seminario Teoría de Grupos
THESIS PRE-DEFENSE -- Ranks over pro-p groups
Ponente: Henrique Souza (UAM)Fecha: jueves 12 de marzo de 2026 - 11:30Lugar: Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Resumen:
The concept of a rank function on a ring was introduced to axiomatize the properties that length functions have over Artinian simple rings. In this talk, we will explore foundational questions on the study of the space of rank functions on the completed group algebra of a pro-p group. These questions are: (1) what are canonical families of rank functions one can study on these algebras and (2) how these canonical families interplay. The canonical families we consider can be summarized in four cases: the inner rank of a matrix, defined over any ring, the Ore rank for Noetherian domains, the ranks induced from finite dimensional continuous representations and the ranks coming from the limits of those along a parametrized family of representations.
We will first work with the class of torsion-free virtually free-by-cyclic pro-p groups G, where we prove that the inner rank on the Fp group algebra satisfies the axioms of a rank function and can be approximated by the regular representation of the finite quotients of G. We moreover prove that both of these properties are commensurability invariants among group algebras of torsion-free pro-p groups.
Afterwards, we will focus on complex group algebras of finitely generated subgroups Γ of semisimple complex Lie groups G. We will pose a conjecture relating the ranks coming from the simple representations of G with the von Neumann rank of Γ. We shall also see how these group algebras locally embed in completed Zp group algebras of a uniform pro-p group U and how to approach this conjecture by studying an approximation of the Ore rank function on Zp[[U]] -- which we accomplish when G is a product of copies of SL2.
