Self-similar groups, automatic sequences, and unitriangular representations

Grigorchuk, Rostislav; Leonov, Yuriy; Nekrashevych, Volodymyr; Sushchansky, V.

doi:10.1007/s13373-015-0077-7

Cited by 9 publications

(10 citation statements)

References 44 publications

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“…For more on the Grigorchuk group or similar groups, the reader can also consult, e.g., [33,26,25,36,30]. Note that automata groups appear to be close to morphic or automatic sequences, while automatic groups (see, e.g., [22]) seem to be rather away from these sequences.…”

Section: Final Remarksmentioning

confidence: 99%

Hidden automatic sequences

Allouche¹,

Dekking²,

Queffélec³

2021

Combinatorial Theory

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An automatic sequence is a letter-to-letter coding of a fixed point of a uniform morphism. More generally, morphic sequences are letter-to-letter codings of fixed points of arbitrary morphisms. There are many examples where an, a priori, morphic sequence with a non-uniform morphism happens to be an automatic sequence. An example is the Lysënok morphism a → aca, b → d, c → b, d → c, the fixed point of which is also a 2-automatic sequence. Such an identification is useful for describing the dynamical systems generated by the fixed point. We give several ways to uncover such hidden automatic sequences, and present many examples. We focus in particular on morphisms associated with Grigorchuk groups.

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Section: Final Remarksmentioning

confidence: 99%

Hidden automatic sequences

Allouche¹,

Dekking²,

Queffélec³

2021

Combinatorial Theory

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“…There are numerous applications of automatic sequences in group theory. For a recent example and a list of further reference we refer the reader to [41]. Consider the automaton A from figure 1.…”

Section: 5mentioning

confidence: 99%

Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution

Grigorchuk¹,

Lenz²,

Nagnibeda³

2017

Groups, Graphs and Random Walks

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There is a recently discovered connection between the spectral theory of Schrödinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth.We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.

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“…We refer the reader to [2] for details. Recent applications of automatic sequences in group theory include [15,19].…”

Section: Introductionmentioning

confidence: 99%

“…Two of the most widely used bases of this space are the Mahler basis and the van der Put bases [28,36]. In the more general settings of the spaces of continuous functions from Z p to a field, several other bases have been used in the literature: Walsh basis [38], Haar basis (used in group theory context, for example, in [6]), Kaloujnine basis [15]. In this paper we will deal with the van der Put basis, which is made of functions χ n (x), n ≥ 0 that are characteristic functions of cylindrical subsets of Z p consisting of all elements that have the p-adic expansion of n as a prefix.…”

Section: Introductionmentioning

confidence: 99%

Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata

Grigorchuk,

Savchuk

2020

Preprint

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The ring Z d of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree T d . Endomorphisms of this tree (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from Z d to itself and automorphisms of T d constitute the group Isom(Z d ). In the case when d = p is prime, Anashin showed in [3] that f ∈ Lip 1 (Z p ) is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of Z p ∩ Q. We generalize this result to arbitrary integer d ≥ 2, describe the explicit connection between the Moore automaton producing such sequence and the Mealy automaton inducing the corresponding endomorphism. Along the process we produce two algorithms allowing to convert the Mealy automaton of an endomorphism to the corresponding Moore automaton generating the sequence of the reduced van der Put coefficients of the induced map on Z d and vice versa. We demonstrate examples of applications of these algorithms for the case when the sequence of coefficients is Thue-Morse sequence, and also for one of the generators of the standard automaton representation of the lamplighter group.

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Self-similar groups, automatic sequences, and unitriangular representations

Cited by 9 publications

References 44 publications

Hidden automatic sequences

Hidden automatic sequences

Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution

Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata

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