Groups that act faithfully on rooted trees can be completed in different ways to obtain profinite groups. The profinite completion of the group maps onto each of them. Determining the kernels of these maps is known as the congruence subgroup problem, by analogy with a similar problem in arithmetic groups. This problem has been studied by various authors over the years, most notably for self-similar groups and (weakly) branch groups. In the case of self-similar regular branch groups, much insight can be gained into this problem using a symbolic-dynamical point of view.
After reviewing some of the known results on the congruence subgroup problem for various combinations of self-similar and (weakly) branch groups, I plan to talk on joint work with Zoran Sunic where we use the symbolic-dynamical approach to describe explicitly one of the kernels of the congruence subgroup problem, for certain self-similar branch groups.
Examples will be given. No previous knowledge of self-similar or branch groups is required.
