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

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.

12 pages; v2: corrected calculations for Basilica group, based on correcting previously published results for this group that turned out to be inaccurate; version for publication

References in corpus (1)

• On the congruence subgroup problem for branch groups

Cited by in corpus (4)

• On maximal subgroups of infinite index in branch and weakly branch groups
• Restricted Hausdorff spectra of $q$-adic automorphisms
• Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups
• $p$-Basilica groups