  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.
