On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

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 $\hat G$ of the groups G. The representations involved have interesting spectral properties investigated in math.GR/9910102. This paper serves as a group-theoretic counterpart to the studies in the mentionned 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.

[1]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[2]  Н. Гупта,et al.  Some infinite p-groups. , 1983 .

[3]  Laurent Bartholdi,et al.  On the Spectrum of Hecke Type Operators related to some Fractal Groups , 1999 .

[4]  Ralph Duncan James,et al.  Proceedings of the International Congress of Mathematicians , 1975 .

[5]  R. Grigorchuk,et al.  The profinite completion of certain torsion p-groups , 2000 .

[6]  Spectra of non-commutative dynamical systems and graphs related to fractal groups , 2000, math/0012174.

[7]  S. Sidki.  On a 2-generated infinite 3-group: The presentation problem , 1987 .

[8]  Laurent Bartholdi,et al.  Lower Bounds on The Growth of a Group Acting on the Binary Rooted Tree , 2001, Int. J. Algebra Comput..

[10]  R. I. Grigorchuk,et al.  Just Infinite Branch Groups , 2000 .

[11]  N. Boston.  p-adic Galois Representations and pro-p Galois Groups , 2000 .

[12]  Narain Gupta,et al.  On the Burnside problem for periodic groups , 1983 .

[13]  S. Sidki.  On a 2-generated infinite 3-group: Subgroups and automorphisms , 1987 .

[14]  Groups St Andrews 1997 in Bath, I: On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata , 1999 .

[15]  L. Bartholdi,et al.  Lower Bounds on the Growth of Grigorchuk ' s Torsion Group , 1998 .

[16]  Hyman Bass,et al.  Cyclic Renormalization and Automorphism Groups of Rooted Trees , 1995 .

[17]  R. Grigorchuk,et al.  Lie Methods in Growth of Groups and Groups of Finite Width , 2000, math/0002010.

[18]  S. Sidki,et al.  A just-nonsolvable torsion-free group defined on the binary tree , 1999 .

[19]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[20]  Narain Gupta,et al.  Some infinitep-groups , 1983 .

[21]  Irving Reiner,et al.  Methods of Representation Theory , 1981 .

[22]  I. Lysenok.  A system of defining relations for a Grigorchuk group , 1985 .

[23]  C. Curtis,et al.  Methods of representation theory--with applications to finite groups and orders , 1981 .

[24]  La Harpe,et al.  Topics in Geometric Group Theory , 2000 .

[25]  R. Grigorchuk,et al.  Bernside's problem on periodic groups , 1980 .

[26]  Dan Segal,et al.  The finite images of finitely generated groups , 2001 .

[27]  P. Harpe,et al.  Constructing irreducible representations of discrete groups , 1997 .

[28]  THE CHARACTER VALUES OF MULTIPLICITY-FREE IRREDUCIBLE CONSTITUENTS OF A TRANSITIVE PERMUTATION REPRESENTATION , 2001 .

[29]  L. Bartholdi.  Lie Algebras and Growth in Branch Groups , 2001, math/0101222.

[30]  G. Mackey.  The theory of unitary group representations , 1976 .

[31]  A. Shalev,et al.  Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras , 1997 .