The equations satisfied by GGS-groups and the abelian group structure of the Gupta-Sidki group

The group Γ of p-adic automorphisms of the p-adic rooted tree can naively be identified with the p-group FpN, using pointwise addition + in the portraits of the automorphisms. We prove that the operation + is also internal for the Gupta-Sidki group, as well as for some other discrete subgroups of Γ. In order to get this, we introduce the notion of equation, or pattern, for subgroups of Γ, and we describe all equations for these groups.