Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms

TL;DR

This paper generalizes the congruence subgroup problem for groups acting on rooted trees by considering pro-$ ext{C}$ completions, providing criteria for when these groups satisfy the $ ext{C}$-congruence subgroup property, with applications to specific groups.

Contribution

It introduces the $ ext{C}$-congruence subgroup property for groups acting on rooted trees and establishes conditions under which these groups satisfy this property, including necessary and sufficient criteria.

Findings

Weakly regular branch groups can have the $ ext{C}$-CSP under certain conditions.

The Basilica group and GGS-groups with constant vectors have the $p$-CSP.

Provides a framework to compare pro-$ ext{C}$ completions with level stabilizer completions.

Abstract

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-$\mathcal{C}$ completions of the group, where $\mathcal{C}$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $\mathcal{C}$-congruence subgroup property ($\mathcal{C}$-CSP) if its pro-$\mathcal{C}$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $\mathcal{C}$-CSP. In the case where $\mathcal{C}$ is also closed under extensions (for instance the class of all finite $p$-groups for some prime $p$), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the $p$-CSP.