The profinite completion of multi-EGS groups (Q2657659)

In this paper, the authors consider the class of multi-EGS groups that generalize the famous Grigorchuk groups and Gupta-Sidki groups acting on rooted trees. A group \(G\) acting on a rooted tree has the congruence subgroup property if every finite index subgroup of \(G\) contains some level stabilizer. The authors of this paper classify the multi-EGS groups which have the congruence subgroup property. Further, they determine the profinite completion of a multi-EGS group without the congruence subgroup property as a subgroup of automorphism group of the tree \(T\) on which it acts. Finally the authors prove that if \(G\) is a branch multi-EGS group acting on a tree \(T\) then the normalizer \(N_{\mathrm{Aut}(T)}(G)=\mathrm{Aut}(G)\).