# On Parabolic Subgroups and Hecke Algebras of some Fractal Groups

Bartholdi, Laurent; Grigorchuk, Rostislav

Serdica Mathematical Journal (2002)

- Volume: 28, Issue: 1, page 47-90
- ISSN: 1310-6600

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topBartholdi, Laurent, and Grigorchuk, Rostislav. "On Parabolic Subgroups and Hecke Algebras of some Fractal Groups." Serdica Mathematical Journal 28.1 (2002): 47-90. <http://eudml.org/doc/11547>.

@article{Bartholdi2002,

abstract = {* The authors thank the “Swiss National Science Foundation” for its support.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.},

author = {Bartholdi, Laurent, Grigorchuk, Rostislav},

journal = {Serdica Mathematical Journal},

keywords = {Branch Group; Fractal Group; Parabolic Subgroup; Quasi-Regular Representation; Hecke Algebra; Gelfand Pair; Growth; L-Presentation; Tree-like Decomposition; branch groups; fractal groups; parabolic subgroups; quasi-regular representations; Hecke algebras; Gelfand pairs; growth; -presentations; tree-like decompositions},

language = {eng},

number = {1},

pages = {47-90},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {On Parabolic Subgroups and Hecke Algebras of some Fractal Groups},

url = {http://eudml.org/doc/11547},

volume = {28},

year = {2002},

}

TY - JOUR

AU - Bartholdi, Laurent

AU - Grigorchuk, Rostislav

TI - On Parabolic Subgroups and Hecke Algebras of some Fractal Groups

JO - Serdica Mathematical Journal

PY - 2002

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 28

IS - 1

SP - 47

EP - 90

AB - * The authors thank the “Swiss National Science Foundation” for its support.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.

LA - eng

KW - Branch Group; Fractal Group; Parabolic Subgroup; Quasi-Regular Representation; Hecke Algebra; Gelfand Pair; Growth; L-Presentation; Tree-like Decomposition; branch groups; fractal groups; parabolic subgroups; quasi-regular representations; Hecke algebras; Gelfand pairs; growth; -presentations; tree-like decompositions

UR - http://eudml.org/doc/11547

ER -

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