On a free group of transformations defined by an automaton
TL;DR: It is proved that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.
Abstract: We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3
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01 Jan 2005
TL;DR: A survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ as mentioned in this paper.
Abstract: This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ.
154 citations
16 Jul 2011
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.
Abstract: This article combines the features of a survey and a research paper. It presents a review of some 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. Special emphasis is placed on the study of essentially free actions of selfsimilar groups, which are antipodes to branch actions. At the same time, the theme “free versus nonfree” runs through the paper. Sufficient conditions are obtained for the essential freeness of an action of a self-similar group on the boundary of a tree. Specific examples of such actions are given. Constructions of the associated dynamical system and the Schreier dynamical system generated by a Schreier graph are presented. For groups acting on trees, a trace on the associated C*-algebra generated by a Koopman representation is introduced, and its role in the study of von Neumann factors, the spectral properties of groups, Schreier graphs, and elements of the associated C*-algebra is demonstrated. The concepts of asymptotic expander and asymptotic Ramanujan graph are introduced, and examples of such graphs are given. Questions related to the notion of the cost of action and the notion of rank gradient are discussed.
93 citations
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TL;DR: Most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet can be found in, with the exception of.
Abstract: This article contains most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet, extending the previous papers 0704.3876 and math/0612178.
44 citations
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.
Abstract: We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.
41 citations
TL;DR: In this article, it was shown that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary of T and a subgroup of G with trivial stabilizer w, or there exists w 2 @T and a free group of G fixing w and acting on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of groups.
Abstract: We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer of w, or there exists w 2 @T and a free subgroup of G fixing w and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki.
41 citations
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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.
Abstract: Basic definitions and examples Algebraic theory Limit spaces Orbispaces Iterated monodromy groups Examples and applications Bibliography Index
520 citations
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.
Abstract: 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 New classes of groups are introduced corresponding to diverse growth functions and circuit structure In the context of automorphisms of the binary tree, we discuss the structure of maximal 2-subgroups and the question of existence of free subgroups Moreover, we construct Burnside 2-groups generated by automorphisms of the binary tree which are finite state, bounded, and acyclic
155 citations
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.
Abstract: We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedonska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.
73 citations
TL;DR: Finite state automata are constructed which show that GL(n, Z) is embeddable in the group of finite state automorphisms of the one-rooted regular tree of valency 2n.
Abstract: The linear group GL(n, Z) is residually finite by virtue of its action on the (one-rooted) regular 2n-ary coset tree for Zn. In this paper we construct finite state automata which effect this action. This shows that GL(n, Z) is embeddable in the group of finite state automorphisms of the one-rooted regular tree of valency 2n.
56 citations
TL;DR: In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet.
Abstract: In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet.
11 citations