Date of Award
Doctor of Philosophy (PhD)
In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel for the only known group for which it is non-trivial, theHanoi towers group. There they determined the kernel was the Klein 4 group. We present a simpler proof of this theorem. In thecourse of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, self-replicating, regular branch group.
We then 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, adding infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.
Skipper, Rachel, "On a Generalization of the Hanoi Towers Group" (2018). Graduate Dissertations and Theses. 60.