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Title: On Parabolic Subgroups and Hecke Algebras of some Fractal Groups
Authors: Bartholdi, Laurent
Grigorchuk, Rostislav
Keywords: Branch Group
Fractal Group
Parabolic Subgroup
Quasi-Regular Representation
Hecke Algebra
Gelfand Pair
Tree-like Decomposition
Issue Date: 2002
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 28, No 1, (2002), 47p-90p
Abstract: 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.
Description: * The authors thank the “Swiss National Science Foundation” for its support.
ISSN: 1310-6600
Appears in Collections:Volume 28 Number 1

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