[Paper Review] On the Spectrum of Hecke Type Operators related to some Fractal Groups

Laurent Bartholdi, Rostislav Grigorchuk|arXiv (Cornell University)|Oct 19, 1999

Mathematical Analysis and Transform Methods16 references19 citations

TL;DR

This paper constructs the first known example of a connected 4-regular graph whose Laplace operator has a spectrum that is a Cantor set, using Schreier graphs of fractal groups of intermediate growth. The spectrum arises as a transformation of Julia sets from quadratic maps, with spectral computations achieved via a finite approximation method exploiting self-similarity and ad hoc factorization of operators.

ABSTRACT

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also ``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined. In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.

Motivation & Objective

• To compute the spectrum of Hecke-type operators associated with quasi-regular representations of fractal groups.
• To investigate the spectral properties of Laplace operators on Schreier graphs of groups with intermediate growth.
• To establish a connection between the spectra of these operators and the Julia sets of quadratic maps.
• To explore the role of self-similarity in spectral theory through finite approximation techniques.
• To address open questions about spectral gaps and representations in nonamenable, torsion-free groups.

Proposed method

• Utilizes Schreier graphs of fractal groups (e.g., Grigorchuk group and its variants) as underlying graphs for spectral analysis.
• Applies a finite approximation method where spectra of finite quotients are computed and then taken to a limit.
• Employs ad hoc factorization processes to compute spectra of finite counterparts of Hecke-type operators.
• Relies on the self-similarity of the groups to reflect in the self-similarity of the approximating operators.
• Connects the spectra to Julia sets of quadratic polynomials via functional transformations.
• Uses the Markov operator associated with noncommutative dynamical systems to define the spectral problem.

Experimental results

Research questions

• RQ1 Under what conditions does the spectrum of a quasi-regular representation ρ_G/P lie within the spectrum of the full representation ρ_G?
• RQ2 When does equality hold: spec(ρ_G/P) = spec(ρ_G)?
• RQ3 Can a finitely generated, non-residually-finite group without free subgroups be constructed via spectral methods?
• RQ4 Are there groups G < Aut(T) such that the Schreier graphs S(G, P_n, S) form a sequence of expanders or Ramanujan graphs?
• RQ5 Does there exist a torsion-free group G < Aut(T) with a totally disconnected or gap-containing spectrum for its Laplace operator?

Key findings

• The paper presents the first example of a connected 4-regular graph whose Laplace operator has a spectrum that is a Cantor set.
• The spectrum of the Hecke-type operator for the group ̃G is shown to be a Cantor set, arising as a transformed Julia set of a quadratic map.
• The finite approximation method successfully computes spectra by factoring the resolvent of finite operators through self-similar structure.
• For the group ̃G, the spectrum of the Laplace operator Δ is fully recovered via the bipartite structure of its Cayley graph.
• The spectrum of the Markov operator on the Schreier graph of the group Γ is shown to be the union of a Cantor set and a countable set, analogous to results in [Mal95] but under different assumptions.
• The authors establish a spectral link between noncommutative dynamical systems and the spectra of Hecke-type operators via unitary representations and Radon-Nikodým derivatives.