p-Basilica Groups

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 have infinitely many non-normal maximal subgroups.

On the Basilica operation

“…We explicitly compute the Hausdorff dimension of Bas The above equality agrees with the formula of the Hausdorff dimension of p-Basilica groups given by [13], and also with the Hausdorff dimension of the original Basilica group B given in [4]. where the data Y , Q, R and ˆare specified in Section 6.…”

“…The group G has the p-CSP if every subgroup of index a power of p in G contains some layer stabiliser in G. In [18] one finds a sufficient condition for a weakly branch group to have the p-CSP and it is also proved that the original Basilica group B has the 2-CSP. This argument is general-ised by Di Domenico, Fernández-Alcober, Noce and Thillaisundaram [13] to see that the p-Basilica groups have the p-CSP. We further generalise these result.…”

“…Here we make use of Theorem 1.4 to obtain a normal generating set for the layer stabilisers of the generalised Basilica groups (Theorem 5.1). We remark that the result of [13] on p-Basilica groups can be generalised to all d 2 with additional work.…”

“…In the second half of this paper we study the class of generalised Basilica groups Bas s .O d m /, for d; m; s 2 N C with m; s 2, defined by applying Bas s to the free abelian group of rank d with a self-similar action derived from the m-adic odometer. We remark that the above generalisation of the original Basilica group B is different from the one given in [8], but it includes the class of p-Basilica groups, where p is a prime, studied recently in [13]. For every odd prime p, we obtain the p-Basilica group by setting d D 1, m D p and s D 2 in Bas s .O d m /.…”