On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.