## ULethbridge Number Theory and Combinatorics Seminar: Oussama Hamza

- Date: 01/30/2023
- Time: 11:00

University of Lethbridge

Filtrations, arithmetic and explicit examples in an equivariant context

Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n

such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields.

In 2016, Minac, Rogelstad and Tan inferred an explicit relation between (a_n) and (c_n). Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

**Location**: Online

**Time: **12pm Mountain/ 11am Pacific

Live access links are posted here and on https://researchseminars.org/seminar/NTC. For more information, contact Félix Baril Boudreau or Bobby Miraftab.

A recording of this event is available on mathtube.org.