On the Basilica operation

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