p-Basilica Groups
(2022) In Mediterranean Journal of Mathematics 19(6).- Abstract
We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they... (More)
We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.
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- author
- Di Domenico, Elena ; Fernández-Alcober, Gustavo A. ; Noce, Marialaura and Thillaisundaram, Anitha LU
- organization
- publishing date
- 2022
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- congruence subgroup properties, Groups acting on rooted trees, Hausdorff dimension, maximal subgroups, weakly branch groups
- in
- Mediterranean Journal of Mathematics
- volume
- 19
- issue
- 6
- article number
- 275
- publisher
- Birkhäuser Verlag
- external identifiers
-
- scopus:85140997954
- ISSN
- 1660-5446
- DOI
- 10.1007/s00009-022-02187-z
- language
- English
- LU publication?
- yes
- id
- a5e15d5f-255f-41d8-b987-31868f7b2fd3
- date added to LUP
- 2022-12-05 13:52:37
- date last changed
- 2022-12-05 17:00:38
@article{a5e15d5f-255f-41d8-b987-31868f7b2fd3,
abstract = {{<p>We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.</p>}},
author = {{Di Domenico, Elena and Fernández-Alcober, Gustavo A. and Noce, Marialaura and Thillaisundaram, Anitha}},
issn = {{1660-5446}},
keywords = {{congruence subgroup properties; Groups acting on rooted trees; Hausdorff dimension; maximal subgroups; weakly branch groups}},
language = {{eng}},
number = {{6}},
publisher = {{Birkhäuser Verlag}},
series = {{Mediterranean Journal of Mathematics}},
title = {{p-Basilica Groups}},
url = {{http://dx.doi.org/10.1007/s00009-022-02187-z}},
doi = {{10.1007/s00009-022-02187-z}},
volume = {{19}},
year = {{2022}},
}