Weakly branch actions: first-order theory, rigidity and Boston's conjecture

We disprove a well-known conjecture of Boston (2000), which claims that a just-infinite pro-$p$ group is branch if and only if it admits a positive-dimensional embedding in the group of $p$-adic automorphisms. This is obtained as a result of a comprehensive study of the rigidity of branch actions. Firstly, we generalize the notion of the structure graph, introduced by Wilson in 2000, to weakly branch groups and use it to prove several results on the first-order theory of weakly branch groups, extending previous results of Wilson on branch groups. Secondly, we completely characterize the rigidity of weakly branch and branch actions on arbitrary spherically homogeneous rooted trees, extending previous partial results (for branch actions) by Hardy, Garrido, Grigorchuk and Wilson. Moreover, we prove that rigidity of a weakly branch group is equivalent to rigidity of its closure in the full automorphism group. Thirdly, we extend greatly the sufficient conditions $(*)$ and $(**)$ of Grigorchuk and Wilson, which leads to a complete and very easy-to-check characterization of the rigidity of the weakly branch actions of a fractal group of $p$-adic automorphisms. We further establish the first connection in the literature between the Hausdorff dimension of a weakly branch action and its rigidity. Lastly, we put everything together to show that the zero-dimensional just-infinite branch pro-$p$ groups introduced recently by the author admit rigid branch actions on a tree obtained by deletion of levels. This, together with previous results of the author, shows that these groups are indeed counterexamples to the aforementioned conjecture of Boston.