  A theoretical framework is established for explicitly calculating rigid
kernels of self-similar regular branch groups. This is applied to a new
infinite family of branch groups in order to provide the first examples of
self-similar, branch groups with infinite rigid kernel. The groups are analogs
of the Hanoi Towers group on 3 pegs, based on the standard actions of finite
dihedral groups on regular polygons with odd numbers of vertices, and the rigid
kernel is an infinite Cartesian power of the cyclic group of order 2, except
for the original Hanoi group. The proofs rely on a symbolic-dynamical approach,
related to finitely constrained groups.
