Oussama Hamza

During the 60's, Jennings, Golod, Shafarevich and Lazard introduced two sequences of integers $(a_n)$ and $(c_n)$, closely related to a special filtration of a finitely generated pro-$p$ group $G$, called Zassenhaus filtration. These sequences give the cardinality of $G$, and characterize its topology. Let us cite the famous Gocha's alternative: this is a condition on $(a_n)$ and $(c_n)$ equivalent for G to be analytic, i.e a Lie group over $p$-adic fields. Recently, in 2016, Minac, Rogelstad and Tan inferred an explicit relation between $(a_n)$ and $(c_n)$.

 

This talk will review these results, then considering geometrical ideas of Filip and Stix, we enrich them in an isotypical context, i.e when there exists a finite group $\Delta \subset Aut(G)$ of order a prime $q$ dividing $p-1$.