On the congruence subgroup problem for branch groups

For any infinite group with a distinguished family of normal subgroups of finite index -- congruence subgroups-- one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for $SL(n,\mathbb{Z})$ where $n\geq 3$, by Mennicke and Bass, Lazard and Serre. \\

Groups acting on infinite rooted trees are a natural setting in which to ask this question. In particular, branch groups have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.