$p$-Basilica groups

TL;DR

This paper introduces the $p$-Basilica groups acting on $p$-adic trees, exploring their subgroup properties, fractality, and Hausdorff dimensions, revealing novel examples in the theory of branch groups.

Contribution

It provides the first examples of weakly branch groups with the $p$-congruence property but not the full congruence properties, and introduces super strongly fractal, weakly branch groups that are not branch.

Findings

$p$-Basilica groups have the $p$-congruence subgroup property.

They lack the congruence and weak congruence subgroup properties.

These groups are super strongly fractal but not branch.

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.