Title: On the dimension of the set of commutators in weakly branch groups of $p$-adic automorphisms.

Abstract: Let $G$ be a weakly branch subgroup of the group of automorphisms of a $p$-adic tree $T$, for $p$ a prime. We study the set of commutators of $G$ from the perspective of its box dimension inside the derived subgroup $G'$, where $G'$ is considered as a metric space with respect to the distance induced by its congruence subgroups. We prove that if $G$ consists of $p$-adic automorphisms, i.e. if $G$ is contained in a Sylow pro-$p$ subgroup of the group of automorphisms of $T$, then this box dimension is always $1$.