On Parabolic Subgroups and Hecke Algebras of some Fractal Groups

Bartholdi, 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 -