GGS-groups acting on trees of growing degrees

TL;DR

This paper introduces growing GGS-groups acting on trees with increasing degrees, revealing new properties such as branch structures, the p-congruence subgroup property, and the existence of Beauville groups, expanding understanding of infinite branch groups.

Contribution

It provides the first examples of finitely generated branch groups with infinitely many finite-index maximal subgroups and explores their algebraic properties.

Findings

Existence of growing GGS-groups with branch and p-congruence subgroup properties.

Construction of groups with only finite index maximal subgroups.

Congruence quotients form Beauville groups.

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.