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

TL;DR

This paper investigates the self-similarity index of $p$-adic analytic pro-$p$ groups, showing that for primes $p \\geq 3$, there are infinitely many such groups with arbitrarily large self-similarity indices, regardless of their dimension.

Contribution

It demonstrates the unbounded nature of the self-similarity index in $p$-adic analytic pro-$p$ groups for primes $p \\geq 3$, highlighting the diversity of these groups.

Findings

Existence of infinitely many non-isomorphic groups with large self-similarity index.

Self-similarity index cannot be bounded solely by the group's dimension.

Results hold for all integers $d$ and primes $p \\geq 3$.

Abstract

Let $p$ be a prime. We say that a pro-$p$ group is self-similar of index $p^k$ if it admits a faithful self-similar action on a $p^k$-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 $p^k$, such that $G$ is self-similar of index $p^k$. We show that for every prime $p\geqslant 3$ 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.