On the Basilica operation

Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G ď AutpT q of a rooted tree T a family of Basilica groups BasspGq, s P N`. For the dyadic odometer O2, one has B “ Bas2pO2q. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling BasspGq, in case G fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain GGS-groups and of generalisations of the odometer, O m. Furthermore, we study the structure of groups of type BasspO mq and prove an analogue of the congruence subgroup property in the case m “ p, a prime.