Harmonic Analysis on Free Groups

Figá-Talamanca, Alessandro; Picardello, Massimo A.

doi:10.1201/9781003071747

Cited by 57 publications

(29 citation statements)

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“…Another important case is trees and tree like graphs, such as, the Cayley graphs of free groups and free product of finite groups. For these groups the use of representation theory is limited but somehow possible, the structure of the spectrum is similar to the case of graphs with co-compact

-action, although the methods are quite different, see [ 86 , 88 , 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 ].…”

Section: Spectra Of Groups and Graphsmentioning

confidence: 99%

Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

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Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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-action, although the methods are quite different, see [ 86 , 88 , 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 ].…”

Section: Spectra Of Groups and Graphsmentioning

confidence: 99%

Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

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“…The Helgason-Fourier transform can be defined on homogeneous trees (see [8,10,11]) in analogy with the setup of symmetric spaces [14]. We briefly recall its definition and its main features.…”

Section: The Helgason-fourier Transform On Homogeneous Treesmentioning

confidence: 99%

“…As the Euclidean Fourier transform, the Helgason-Fourier transform extends to a unitary operator on L 2 (X ) (see [10,11]). The Plancherel measure involves a version of the Harish-Chandra c-function inspired by the symmetric space construction [12], namely the meromorphic function…”

Section: Definition 3 the Helgason-fourier Transform Of F ∈ C C (X ) With Respect To The Vertexmentioning

confidence: 99%

“…By Proposition 9, F ∈ L 2 ( ) implies that v F satisfies (9) for every v ∈ X . Conversely, if we want to prove that a function F ∈ L 2 ( ) satisfies (11) it is enough to verify that (9) holds true for at least one, hence every, v ∈ X . This last remark will prove very useful in our proofs.…”

Section: Proposition 9 Letmentioning

confidence: 99%

“…The horocyclic Radon transform on homogeneous trees was introducted by Cartier [5] and studied by Figà-Talamanca and Picardello [11], Betori et al [3], Cowling et al Setti [8], Casadio Tarabusi et al [6], and Veca [17], to name a few. Some of the typical issues considered are inversion formulae and range problems.…”

Section: Introductionmentioning

confidence: 99%

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