[Paper Review] On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

Laurent Bartholdi, Rostislav Grigorchuk|arXiv (Cornell University)|Nov 25, 1999

Mathematical Dynamics and Fractals6 references16 citations

TL;DR

This paper introduces parabolic subgroups in fractal branch groups, proving they are weakly maximal and yield irreducible quasi-regular representations that are approximated by finite-dimensional ones. The key contribution is showing the associated Hecke algebras are commutative, enabling complete decomposition of finite quasi-regular representations via double cosets, and constructing the first example of a torsion-free branch just-infinite group with intermediate growth.

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 $\hat G$ 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.

Motivation & Objective

• To investigate the subgroup structure of fractal groups of branch type, particularly parabolic subgroups arising as stabilizers of infinite paths in rooted trees.
• To analyze the quasi-regular representations associated with these parabolic subgroups and determine their irreducibility and approximation by finite-dimensional representations.
• To study the structure of Hecke algebras attached to parabolic subgroups and establish their commutativity, enabling decomposition of finite quasi-regular representations.
• To construct new examples of branch groups, including the first torsion-free branch just-infinite group and a new example of intermediate growth.
• To extend results to profinite completions of the groups and establish connections to spectral properties and Gelfand pairs.

Proposed method

• Define parabolic subgroups as stabilizers of boundary points or infinite geodesic paths in the regular rooted tree on which the group acts.
• Use the inverse limit structure $P = \bigcap_{n} P_n$, where $P_n$ is the stabilizer of a finite path of length $n$, to approximate the infinite-dimensional quasi-regular representation $\rho_{G/P}$ by finite-dimensional ones $\rho_{G/P_n}$.
• Prove that the Hecke algebra $\mathcal{L}(G, P_n)$ is abelian by showing it is a direct summand of the next algebra and that the difference in dimension is small ($d-1 < 4$), implying semi-simplicity and commutativity.
• Apply the Gelfand pair criterion: since $\mathcal{L}(G, P_n)$ is abelian, the pair $(G, P_n)$ is a Gelfand pair, ensuring all irreducible components in $\rho_{G/P_n}$ have multiplicity one.
• Use the number of double cosets $P_n g P_n$ to compute the dimension of the Hecke algebra and thus the number of irreducible components in $\rho_{G/P_n}$.
• Construct explicit presentations for examples, including an $L$-type presentation for a new branch group of intermediate growth, and verify orbit structures via induction on $n$.

Experimental results

Research questions

• RQ1 Are the parabolic subgroups of fractal branch groups weakly maximal, and do they yield irreducible quasi-regular representations?
• RQ2 What is the structure of the Hecke algebra $\mathcal{L}(G, P_n)$ for parabolic subgroups, and is it commutative?
• RQ3 Can the finite-dimensional quasi-regular representations $\rho_{G/P_n}$ be completely decomposed into irreducible components, and how is this related to double cosets?
• RQ4 Does there exist a torsion-free branch just-infinite group, and can such a group have intermediate growth?
• RQ5 How do the spectral properties of the representations relate to fractal geometry, particularly Julia sets?

Key findings

• The parabolic subgroups $P$ of fractal branch groups are weakly maximal and the corresponding quasi-regular representations $\rho_{G/P}$ are irreducible.
• The Hecke algebra $\mathcal{L}(G, P_n)$ is abelian for all $n$, which implies that the decomposition of $\rho_{G/P_n}$ into irreducible components is multiplicity-free and determined by the number of double cosets $P_n g P_n$.
• The number of irreducible components in $\rho_{G/P_n}$ is equal to the dimension of $\mathcal{L}(G, P_n)$, which is computed as $2n+1$ for the group $\overline{\Gamma}$, corresponding to the orbits of $P_n$ on $\Sigma^n$.
• The representation $\rho_{G/P_n}$ decomposes as $\rho_{G/P_0} \oplus \bigoplus_{i=1}^{d-1} A_{n,i}$, where each $A_{n,i}$ is an irreducible representation of dimension $d^{n-1}$, with $d$ the degree of the tree.
• The paper constructs the first example of a torsion-free branch just-infinite group, providing a new class of groups with intermediate growth and a new $L$-type presentation.
• The results extend to the profinite completion $\widehat{G}$, and the finite-dimensional approximations $\rho_{G/P_n}$ are used to study spectral properties related to Julia sets of quadratic maps.