On the Basilica operation

Abstract

Inspired by the Basilica group $\mathcal{B}$, we describe a general construction which allows us to associate to any group of automorphisms $G\le \mathrm{Aut}(T)$ of a rooted tree $T$ a family of Basilica groups ${\mathrm{Bas}}_s(G)$, $s\in {\mathbb{N}}_{+}$. For the dyadic odometer ${\mathcal{O}}_2$, one has $\mathcal{B}={\mathrm{Bas}}_2({\mathcal{O}}_2)$. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling ${\mathrm{Bas}}_s(G)$, in case $G$ fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain $\mathsf{GGS}$-groups and of generalisations of the odometer, ${\mathcal{O}}_m^d$. Furthermore, we study the structure of groups of type ${\mathrm{Bas}}_s({\mathcal{O}}_m^d)$ and prove an analogue of the congruence subgroup property in the case $m=p$, a prime.