Ergodic Decomposition of Group Actions on Rooted Trees

Grigorchuk, Rostislav; Savchuk, Dmytro

doi:10.1134/s0371968516010064

Cited by 3 publications

(2 citation statements)

References 29 publications

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“…For example, Grigorchuk group was the first example of a group of intermediate growth [Gri83], as well as one of the first examples of infinite finitely generated torsion groups [Gri80] (see also [Ale72,Sus79,GS83]). Now this is a rich theory with connections to combinatorics [G Š06], analysis [GLS Ż00], holomorphic dynamics [Nek05], dynamical systems [GS16], computer science [MS15], and many other areas. We refer the reader to the survey article [GNS00] for history and references.…”

Section: Introductionmentioning

confidence: 99%

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Ahmed

Savchuk

2019

J. Algebra Appl.

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We show that every polynomial in Z[x] defines an endomorphism of the dary rooted tree induced by its action on the ring Z d of d-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in Z[x] of the same degree. In the case of permutational polynomials acting on Z d by bijections the induced endomorphisms are automorphisms of the tree. In the case of Z 2 such polynomials were completely characterized by Rivest in [Riv01]. As our main application we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial f (x) ∈ Z[x] that is necessary and sufficient for f to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of f (x) on Z 2 with respect to the normalized Haar measure.

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Section: Introductionmentioning

confidence: 99%

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Ahmed

Savchuk

2019

J. Algebra Appl.

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“…Consider the tree with vertices the connected components of the powers of A, and the incidence relation built by adding an element of Q: for any n ≥ 0, the connected component of u ∈ Q n is linked to the connected component(s) of ux, for any x ∈ Q. This tree is called the orbit tree of d(A) [4,10]. It can be seen as the quotient of the tree Q * under the action of the group d(A) .…”

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confidence: 99%

Connected reversible Mealy automata of prime size cannot generate infinite Burnside groups

Godin,

Klimann

2016

Preprint

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The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most 3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to automata with bigger stateset, proving that, if a connected reversible automaton has a prime number of states, it cannot generate an infinite Burnside group.

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On bireversible Mealy automata and the Burnside problem

Godin

Klimann

2018

Theoretical Computer Science

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Ergodic Decomposition of Group Actions on Rooted Trees

Cited by 3 publications

References 29 publications

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Connected reversible Mealy automata of prime size cannot generate infinite Burnside groups

On bireversible Mealy automata and the Burnside problem

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