Geodesic
On Parabolic Subgroups and Hecke Algebras of some Fractal Groups
Serdica Mathematical Journal, Tome 28 (2002) no. 1, pp. 47-90.

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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 [6]. 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 relators.
Keywords: Branch Group, Fractal Group, Parabolic Subgroup, Quasi-Regular Representation, Hecke Algebra, Gelfand Pair, Growth, L-Presentation, Tree-like Decomposition 
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Bartholdi, Laurent; Grigorchuk, Rostislav. On Parabolic Subgroups and Hecke Algebras of some Fractal Groups. Serdica Mathematical Journal, Tome 28 (2002) no. 1, pp. 47-90. http://geodesic.mathdoc.fr/item/SMJ2_2002_28_1_a2/