The congruence subgroup problem for a family of branch groups

We construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel. The only previously known group where this kernel is non-trivial is the Hanoi towers group and so this adds infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.

[1]  Michiel Hazewinkel,et al.  Handbook of algebra , 1995 .

[2]  Olivier Siegenthaler.  Discrete and Profinite Groups Acting on Regular Rooted Trees , 2010 .

[3]  Alejandra Garrido,et al.  On the congruence subgroup property for GGS-groups , 2016, 1604.03465.

[4]  J. Müller,et al.  Group Theory , 2019, Computers, Rigidity, and Moduli.

[5]  L. Bartholdi,et al.  The congruence subgroup problem for branch groups , 2009, 0902.3220.

[6]  Laurent Bartholdi,et al.  The Twisted Twin of the Grigorchuk Group , 2009, Int. J. Algebra Comput..

[7]  R. I. Grigorchuk,et al.  Just Infinite Branch Groups , 2000 .

[8]  Alejandra Garrido.  Abstract commensurability and the Gupta--Sidki group , 2013, 1310.0493.

[9]  T. Hall,et al.  Journal of Algebra , 1964, Nature.

[10]  H. Bateman.  Book Review: Ergebnisse der Mathematik und ihrer Grenzgebiete , 1933 .

[11]  Zoran Sunic,et al.  Groups St Andrews 2005: Self-similarity and branching in group theory , 2007 .

[12]  H. Lenstra.  Profinite Groups , 2022 .

[13]  Alejandra Garrido,et al.  Multi-GGS Groups have the Congruence Subgroup Property , 2017, Proceedings of the Edinburgh Mathematical Society.

[14]  B. M. Fulk.  MATH , 1992 .

[15]  ScienceDirect,et al.  Comptes rendus. Mathématique , 2002 .

[16]  R. Grigorchuk,et al.  Asymptotic aspects of Schreier graphs and Hanoi Towers groups , 2006 .

[17]  A. Shalev,et al.  Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras , 1997 .

[18]  Balint Virag,et al.  Dimension and randomness in groups acting on rooted trees , 2004 .

[19]  Rachel Skipper.  A constructive proof that the Hanoi towers group has non-trivial rigid kernel , 2016, 1606.00772.

[20]  Olivier Siegenthaler.  Hausdorff dimension of some groups acting on the binary tree , 2006, math/0609237.

[21]  Laurent Bartholdi,et al.  Endomorphic Presentations of Branch Groups , 2022 .