On a series of finite automata defining free transformation groups
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In this paper, the authors introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata, and prove that transformations defined by automata from the first series generate a free non-Abelian group of infinite rank.Abstract:
We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that transformations defined by automata from the first series generate a free non-Abelian group of infinite rank, while automata from the second series give rise to the free product of infinitely many groups of order 2.read more
Citations
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Some topics in the dynamics of group actions on rooted trees
TL;DR: In this article, a review of results obtained during the last decade in problems related to the dynamics of branch and self-similar groups on the boundary of a spherically homogeneous rooted tree and to the combinatorics and asymptotic properties of Schreier graphs associated with a group or with its action is presented.
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Automata over a binary alphabet generating free groups of even rank
TL;DR: It is shown that a free group of every finite rank can be generated by finite automata over a binary alphabet, and free products of cyclic groups of order two are constructed via such automata.
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Automata generating free products of groups of order 2
Dmytro Savchuk,Yaroslav Vorobets +1 more
TL;DR: In this article, a family of automata with n states, n ⩾ 4, acting on a rooted binary tree that generate the free products of cyclic groups of order 2 was constructed.
Journal ArticleDOI
Automaton Semigroups: The Two-state Case
TL;DR: It is proved that semigroups generated by reversible two-state Mealy automata have remarkable growth properties: they are either finite or free and an effective procedure to decide finiteness or freeness of such semig groups is given.
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Groups generated by 3-state automata over a 2-letter alphabet. II
Ievgen Bondarenko,Rostislav Grigorchuk,Rostyslav Kravchenko,Yevgen Muntyan,Volodymyr Nekrashevych,Dmytro Savchuk,Zoran Sunic +6 more
TL;DR: In this article, the authors presented results on the classification of groups generated by 3-state automata over a 2-letter alphabet, including cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups and other interesting self-similar groups.
References
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Book
Self-Similar Groups
TL;DR: In this article, the authors define limit spaces, limit spaces and limit spaces in algebraic theory, and use them to define Iterated Monodromy groups (IMG) groups.
Book ChapterDOI
From fractal groups to fractal sets
Abstract: The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of “renormalization group”, which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. The notions of fractal and multi-fractal, playing an important role in singular geometry, measure theory and holomorphic dynamics, are also related. Self-similarity also appears in the theory of C*-algebras (for example in the representation theory of the Cuntz algebras) and in many other branches of mathematics. Starting from 1980 the idea of self-similarity entered algebra and began to exert great influence on asymptotic and geometric group theory.
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Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity
TL;DR: In this paper, a natural interpretation of automorphisms of one-rooted trees as output automata permits the application of notions of growth and circuit structure in their study and new classes of groups are introduced corresponding to diverse growth functions and circuit structures.
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Branch groups
TL;DR: The class of branch groups is defined in this article, both in the abstract and in the profinite category, and the relationship of this class with the class of extremal groups is established.
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Automata and Square Complexes
Yair Glasner,Shahar Mozes +1 more
TL;DR: In this article, a new geometric tool for analyzing groups of finite automata is introduced, which associates a square complex with a product of two trees if the automaton is bi-reversible.