We study the subgroup structure, Hecke algebras, quasi-regular
representations, and asymptotic properties of some fractal groups of branch
type. We introduce parabolic subgroups, show that they are weakly maximal, and
that the corresponding quasi-regular representations are irreducible. These
(infinite-dimensional) representations are approximated by finite-dimensional
quasi-regular representations. The Hecke algebras associated to these parabolic
subgroups are commutative, so the decomposition in irreducible components of
the finite quasi-regular representations is given by the double cosets of the
parabolic subgroup. Since our results derive from considerations on
finite-index subgroups, they also hold for the profinite completions G^
of the groups G. The representations involved have interesting spectral
properties investigated in math.GR/9910102. This paper serves as a
group-theoretic counterpart to the studies in the mentionned paper. We study
more carefully a few examples of fractal groups, and in doing so exhibit the
first example of a torsion-free branch just-infinite group. We also produce a
new example of branch just-infinite group of intermediate growth, and provide
for it an L-type presentation by generators and relators.Comment: complement to math.GR/991010
