Branch Group Fractal Group Parabolic Subgroup Quasi-Regular Representation Hecke Algebra Gelfand Pair Growth L-Presentation Tree-like Decomposition
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 28, No 1, (2002), 47p-90p
We study the subgroup structure, Hecke algebras, quasi-regular
representations, and asymptotic properties of some fractal groups of branch
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
. This paper serves as a group-theoretic counterpart to the
studies in the mentioned 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
* The authors thank the “Swiss National Science Foundation” for its support.