Each GGS-group on the p-adic tree is defined by a
{{(p-1)}}
-tuple of integers modulo p, called the defining tuple. It is proved that two GGS-groups are conjugate (equivalently, by a theorem of Grigorchuk and Wilson, isomorphic) in the automorphism group of the tree if and only if their defining tuples can be transformed into each other by multiplication by an integer modulo p and performing certain reorderings of the tuple. Finally, the number of isomorphism classes of GGS-groups is calculated.
A constant spinal group is a subgroup of the automorphism group of a regular rooted tree, generated by a group of rooted automorphisms A and a group of directed automorphisms B whose action on a subtree is equal to the global action. We provide two conditions in terms of certain dynamical systems determined by A and B for constant spinal groups to be periodic, generalising previous results on Grigorchuk-Gupta-Sidki groups and other related constructions. This allows us to provide various new examples of finitely generated infinite periodic groups.
Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G Ä Aut.T / of a rooted tree T a family of Basilica groups Bas s .G/, s 2 N C . For the dyadic odometer O 2 , one has B D Bas 2 .O 2 /. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling 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 GGS-groups and of generalisations of the odometer, O d m . Furthermore, we study the structure of groups of type Bas s .O d m / and prove an analogue of the congruence subgroup property in the case m D p, a prime.
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