On the Basilica Operation

TL;DR

This paper introduces a new construction linking automorphism groups of rooted trees to Basilica groups, analyzing their properties, dimensions, and structural features, including a congruence subgroup property for certain cases.

Contribution

It generalizes the Basilica group construction to a broad class of automorphism groups, providing tools to analyze their properties and dimensions, and establishing new structural results.

Findings

Calculated Hausdorff dimensions for certain Basilica groups

Identified properties preserved under the Basilica operation

Proved an analogue of the congruence subgroup property for prime cases

Abstract

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