Benjamin Steinberg, Mariya Vorobets, Yaroslav Vorobets
We construct automata over a binary alphabet with $2n$ states, $n\geq 2$, 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.
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For every natural number $n$, we classify abelian groups generated by an $n$-state time-varying automaton over the binary alphabet, as well as by an $n$-state Mealy automaton over the binary alphabet.
Generalizing the idea of self-similar groups defined by Mealy automata, we itroduce the notion of a self-similar automaton and a self-similar group over a changing alphabet. We show that every finitely generated residually-finite group is self-similar over an arbitrary unbounded changing alphabet. We construct some naturally defined self-similar automaton representations over an unbounded changing alphabet for any lamplighter group $K\wr \mathbb{Z}$ with an arbitrary finitely generated (finite or infinite) abelian group $K$.
In the paper, we deal with the notion of an automaton over a changing alphabet, which generalizes the concept of a Mealy-type automaton. We modify the methods based on the idea of a dual automaton and its action used by B. Steinberg et al. (2011) and M. Vorobets and Ya. Vorobets (2007,2010) and adapt them to automata over a changing alphabet. We show that this modification provides some naturally defined automaton representations of a free nonabelian group by a 2-state automaton over a changing alphabet.
Classification of groups generated by 3-state automata over a 2-letter alphabet started in the first paper (see http://www.arxiv.org/abs/math/0612178) is continued.
The notion of an automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ is used to define and study automorphism groups of the tree $X^*$ of finite words over $X$. The concept of bi-reversibility for Mealy-type automata is extended to automata over a changing alphabet. It is proved that a non-abelian free group can be generated by a two-state bi-reversible automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ if and only if $X$ is unbounded. The characterization of groups generated by a two-state bi-reversible automaton over the sequence of binary alphabets is established.
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.
An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite, abelian, and free groups are classified. In addition, we provide detailed information and complete proofs for several groups from the class, with the intention of showing the main methods and techniques used in the classification.
We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite auto…
We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.
We construct the groups $<A,B,C \;| \, A^2,B^2,C^2,(ABC)^2>$ and $<A,B \;| \, A^2,B^4,(AB)^4>$, using 3-state automata over the alphabets $\{1,2,3\}$ and $\{1,2,3,4\}$. In addition, we show, how to define direct powers of $G$ by automaton (when for $G$ it's given), keeping the alphabet.
We give a new example of an automata group of intermediate growth. It is generated by an automaton with 4 states on an alphabet with 8 letters. This automata group has exponential activity and its limit space is not simply connected.
Universal groups of cellular automata
本研究证明任意字母表上的可逆元胞自动机（RCA）群存在由三个对合生成的通用子群，含所有有限生成RCA群的同构副本，并刻画了不同阶字母表下相关RCA群的结构性质。
We add to the classification of groups generated by 3-state automata over a 2 letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic.
A Reversibility Characterization of Locally Finite Groups by Cellular Automata
证明群G是局部有限的当且仅当在每个字母表上，每个双射元胞自动机A^G→A^G都是可逆的；对于非局部有限群，存在无限字母表上的双射元胞自动机其逆不是元胞自动机。
For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.
We prove that a group $G$ is locally finite if and only if every surjective real (or complex) linear cellular automaton with finite-dimensional alphabet over $G$ is injective.
On (bi)reversible automata generating lamplighter groups
本文对任意非平凡阿贝尔群构造可逆（奇数阶时为双可逆）自动机，使其生成群为受限圈积，即灯夫群。
Closed Image Characterizations of Locally Finite Groups via Cellular Automata
证明群G是局部有限的当且仅当每个线性细胞自动机V^G→V^G在prodiscrete拓扑下具有闭图像，解决了Ceccherini-Silberstein和Coornaert的开放问题7。
This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
Graph and wreath products of cellular automata
本文证明了双侧全移位自同构群的子群在可数图积下封闭，引入无 $A$-消除的群作用概念，并证明有限阿贝尔群与元胞自动机群的圈积可嵌入全移位自同构群，同时将结论推广至单侧情形。
On Gottschalk's surjunctivity conjecture for non-uniform cellular automata
本文研究了非均匀细胞自动机在surjunctive群宇宙中的surjunctivity性质，证明了可逆的非均匀细胞自动机必须可逆，并扩展了Capobianco等人关于细胞自动机的对偶surjunctivity定理。
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian…
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