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On the Spectrum of Hecke Type Operators related to some Fractal Groups

Contributors

Bartholdi, Laurent; Grigorchuk, Rostislav

Published in

Proceedings of the Steklov Institute of Mathematics, vol. 231, p. 1-41

Publication date

2000

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.

Classification

  • arxiv : math.GR

UNIGE affiliation entities

Citation (ISO format)

BARTHOLDI, Laurent, GRIGORCHUK, Rostislav. On the Spectrum of Hecke Type Operators related to some Fractal Groups. In: Proceedings of the Steklov Institute of Mathematics, 2000, vol. 231, p. 1–41.

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Article

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Journal ISSN

0081-5438
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Technical informations

Creation

09/11/2010 15:37:00

First validation

09/11/2010 15:37:00

Update

14/03/2023 16:09:10

Status update

14/03/2023 16:09:10

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