On the Basilica Operation
Jan Moritz Petschick, Karthika Rajeev

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 , we describe a general construction which allows us to associate to any group of automorphisms of a rooted tree a family of Basilica groups . For the dyadic odometer , one has . We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling , in case fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain -groups and of generalisations of the odometer, . Furthermore, we study the structure of groups of type and prove an analogue of the congruence subgroup…
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