GGS-groups acting on trees of growing degrees

Abstract

We consider analogues of Grigorchuk-Gupta-Sidki (GGS-)groups acting on trees of growing degree; the so-called growing GGS-groups. These groups are not just infinite and do not possess the congruence subgroup property, but many of them are branch and have the $p$-congruence subgroup property, for a prime $p$. Among them, we find groups with maximal subgroups only of finite index, and with infinitely many such maximal subgroups. These give the first examples of finitely generated branch groups with infinitely many finite-index maximal subgroups. Additionally, we prove that congruence quotients of growing GGS-groups associated to a defining vector of zero sum give rise to Beauville groups.

arXiv comment28 pages; revised version with an added Section 3.4 on the Hausdorff dimension. This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Algebra on 10th July 2025, available at: https://doi.org/10.1080/00927872.2025.2525388