Pro-C congruence properties for groups of rooted tree automorphisms

Abstract

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-C completions of the group, where C is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the C-congruence subgroup property (C-CSP) if its pro-C completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the C-CSP. In the case where C is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.