GGS-groups over primary trees: branch structures

We study branch structures in Grigorchuk–Gupta–Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree p n for a prime p . Apart from a small set of exceptions for p = 2 , we prove that all these groups are weakly regular branch over G ′ ′ . Furthermore, in most cases they are actually regular branch over γ 3 ( G ) . This is a significant extension of previously known results regarding periodic GGS-groups over primary trees and general GGS-groups in the case n = 1 . We also show that, as in the case n = 1 , a GGS-group generated by a constant vector is not branch.