Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

D'Angeli, Daniele; Bondarenko, Ievgen; Nagnibeda, Tatiana

doi:10.4171/jfg/55

Cited by 12 publications

(14 citation statements)

References 23 publications

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“…In this case

is a set of left cosets (for the left version of definition) and

. Again, one can consider a right version of the definition, oriented or non-oriented, labeled or unlabeled versions of the Schreier graph [ 61 , 62 , 63 ].…”

Section: Graphs Of Algebraic Origin and Their Growthmentioning

confidence: 99%

“…A Schreier graph Γ = Γ(G, H, A) is determined by a triple (G, H, A), where as before A is a system of generators of G and H is a subgroup of G. In this case V = {gH | g ∈ G} is a set of left cosets (for the left version of definition) and E = {(gH, agH) | g ∈ G, a ∈ A ∪ A −1 }. Again, one can consider a right version of the definition, oriented or nonoriented, labeled or unlabeled versions of the Schreier graph [61][62][63].…”

Section: Graphs Of Algebraic Origin and Their Growthmentioning

confidence: 99%

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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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“…In this case

is a set of left cosets (for the left version of definition) and

. Again, one can consider a right version of the definition, oriented or non-oriented, labeled or unlabeled versions of the Schreier graph [ 61 , 62 , 63 ].…”

Section: Graphs Of Algebraic Origin and Their Growthmentioning

confidence: 99%

Section: Graphs Of Algebraic Origin and Their Growthmentioning

confidence: 99%

Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

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“…Finite and infinite Schreier graphs have been investigated from a combinatorial point of view in several contexts (e.g., [10,13]). Classifications of infinite Schreier graphs have been studied in several papers (see [4,5,8,9,21] for further discussions about this topic). In this setting, another problem is of considerable interest: the study of the spectral properties of Schreier graphs associated with an automaton group.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 4.17. Theorem 4.16 can be directly proven by using the techniques developed in [5]. Now we pass to the study of isomorphism classes for the infinite Schreier graphs…”

mentioning

confidence: 99%

On an uncountable family of graphs whose spectrum is a Cantor set

Cavaleri¹,

D'Angeli²,

Donno³

et al. 2021

Preprint

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For each p ≥ 1, the star automaton group G Sp is an automaton group which can be defined starting from a star graph on p + 1 vertices. We study Schreier graphs associated with the action of the group G Sp on the regular rooted tree T p+1 of degree p + 1 and on its boundary ∂T p+1 . With the transitive action on the n-th level of T p+1 is associated a finite Schreier graph Γ p n , whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as limits of the sequence {Γ p n } n≥1 in the Gromov-Hausdorff topology. We obtain an explicit description of the spectrum of the graphs {Γ p n } n≥1 . Then, by using amenability of G Sp , we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map f p (z) = z 2 −2(p−1)z −2p, and a countable collection of isolated points supporting the KNS spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have 1, 2 or 2p ends, and that the case of 1 end is generic with respect to the uniform measure on ∂T p+1 .

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“…It has proven to have deep connections with the theory of profinite groups and with complex dynamics. In particular, many groups of this type satisfy a property of self-similarity, reflected on fractalness of some limit objects associated with them [2,5,19]. In the spirit of the modern theory of dynamical systems, one is interested in the action of an automaton group (which is a countable group) on the uncountable set of right infinite sequences somehow endowed with the uniform measure.…”

Section: Introductionmentioning

confidence: 99%

Freeness of automaton groups vs boundary dynamics

D'Angeli

Rodaro

2016

Journal of Algebra

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We prove that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group. We specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to the existence of a finite Schreier graph in the boundary of the enriched dual pointed at some essentially non-trivial point. From these results we derive some consequences from the algebraic, algorithmic and dynamical point of view.

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Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Cited by 12 publications

References 23 publications

Integrable and Chaotic Systems Associated with Fractal Groups

Integrable and Chaotic Systems Associated with Fractal Groups

On an uncountable family of graphs whose spectrum is a Cantor set

Freeness of automaton groups vs boundary dynamics

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