Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G≤Aut(T) of a rooted tree T a family of Basilica groups Bass(G), s∈N+. For the dyadic odometer O2, one has B=Bas2(O2). We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling Bass(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, Omd. Furthermore, we study the structure of groups of type Bass(Omd) and prove an analogue of the congruence subgroup property in the case m=p, a prime.
Classification :
20-XX, 68-XX
Mots-clés :
Basilica group, groups acting on rooted trees, automatic groups, spinal groups, Hausdorff dimension, congruence subgroup property, self-similar groups
Affiliations des auteurs :
Jan Moritz Petschick
1
;
Karthika Rajeev
2
1
Heinrich-Heine-Universität, Düsseldorf, Germany
2
Heinrich-Heine-Universität, Düsseldorf; Universität Bielefeld, Germany
Jan Moritz Petschick; Karthika Rajeev. On the Basilica operation. Groups, geometry, and dynamics, Tome 17 (2023) no. 1, pp. 331-384. doi: 10.4171/ggd/702
@article{10_4171_ggd_702,
author = {Jan Moritz Petschick and Karthika Rajeev},
title = {On the {Basilica} operation},
journal = {Groups, geometry, and dynamics},
pages = {331--384},
year = {2023},
volume = {17},
number = {1},
doi = {10.4171/ggd/702},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/702/}
}
TY - JOUR
AU - Jan Moritz Petschick
AU - Karthika Rajeev
TI - On the Basilica operation
JO - Groups, geometry, and dynamics
PY - 2023
SP - 331
EP - 384
VL - 17
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/702/
DO - 10.4171/ggd/702
ID - 10_4171_ggd_702
ER -
%0 Journal Article
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%A Karthika Rajeev
%T On the Basilica operation
%J Groups, geometry, and dynamics
%D 2023
%P 331-384
%V 17
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/702/
%R 10.4171/ggd/702
%F 10_4171_ggd_702
