GGS-groups over primary trees: branch structures

Abstract

AbstractWe study branch structures in Grigorchuk–Gupta–Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $$p^n$$ p n for a prime p. Apart from a small set of exceptions for $$p=2$$ p = 2 , we prove that all these groups are weakly regular branch over $$G''$$ G ′ ′ . Furthermore, in most cases they are actually regular branch over $$\gamma _3(G)$$ 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 $$n=1$$ n = 1 . We also show that, as in the case $$n=1$$ n = 1 , a GGS-group generated by a constant vector is not branch.