Automata generating free products of groups of order 2

关于Coxeter系统反射的生成函数与自动机

On generating functions and automata associated to reflections in Coxeter systems

本文研究Coxeter系统反射的生成函数与自动机，证明反射前缀语言的正则性，并推导出仿射Coxeter群反射Poincaré级数的简单公式。

On a series of finite automata defining free transformation groups

We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that each automaton in the first series defines a free non-Abelian group while each automaton in the second series defines the free product of groups of order 2. Furthermore, these properties are shared by disjoint unions of any number of distinct automata from either series.

Cross-wired lamplighter groups and linearity of automata groups

We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machine and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a cross-wired lamplighter group and does not embed in the wreath product of a finite group with a torsion free group. For a subfamily of such finite step two nilpotent groups, we prove that their associated automata groups are linear.

Automata and Square Complexes

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 reproduce the theorem of Macedonska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.

Stallings automata for free-times-abelian groups: intersections and index

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric s…

On Finite Monoids of Cellular Automata

For any group $G$ and set $A$, a cellular automaton over $G$ and $A$ is a transformation $τ: A^G \to A^G$ defined via a finite neighborhood $S \subseteq G$ (called a memory set of $τ$) and a local function $μ: A^S \to A$. In this paper, we assume that $G$ and $A$ are both finite and study various algebraic properties of the finite monoid $\text{CA}(G,A)$ consisting of all cellular automata over $G$ and $A$. Let $\text{ICA}(G;A)$ be the group of invertible cellular automata over $G$ and $A$. In the first part, using information on the conjugacy classes of subgroups of $G$, we give a detailed de…

Ranks of finite semigroups of one-dimensional cellular automata

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $τ: A^G \to A^G$ (where $G$ is any group and $A$ is any set), the the…

Groups and Semigroups Defined by Colorings of Synchronizing Automata

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the De Bruijn automata into Mealy aut…

元胞自动机的图积与圈积

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。

关于强连通自动机的直积与商

On Direct Product and Quotient of Strongly Connected Automata

本文研究强连通置换自动机与强连通同步自动机的直积，通过定义同余关系构造商自动机，证明直积可分解为两个商自动机的直积。

Cayley automatic representations of wreath products

We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed.

有限半群和自动机的克罗恩-罗兹复杂度c=1的可判定性

Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata

本文研究了有限半群和自动机的克罗恩-罗兹复杂度c=1的可判定性，通过证明亨克尔、罗兹和斯坦伯格2012年的下界结果在使用 profinite 方法和麦卡蒙德1991年和2001年的结果后是精确的，解决了计算复杂度的算法问题。

Measuring cones and other thick subsets in free groups

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, $λ-$measures and Cesaro measure of thick monoids. Also we improve the asymptotic classification of regular subsets in free groups.

元胞自动机的范畴积

Categorical products of cellular automata

该论文研究元胞自动机的两个范畴，证明在$\mathcal{CA}(G)$中构型空间的范畴积为$(A_1\times A_2)^G$，在$\mathcal{GCA}$中弱积为$(A_1\times A_2)^{G_1*G_2}$，并由此定义元胞自动机及其广义版本的积。

使用组合群论和自动机理论随机生成群元素，以及一个硬件示例

Random generation of group elements using combinatorial group theory and automata theory, along with a hardware example

本文提出一种基于组合群论和自动机理论的有限群随机元素生成新方法，应用于对称群时平均交换次数少于现有方法，并提供硬件实现及时间复杂度改进。

A characterization of those automata that structurally generate finite groups

Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit--that is, where every cycle in the underlying directed graph is a sink component--generates a fi- nite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper.

Words in Linear Groups, Random Walks, Automata and P-Recursiveness

Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of Kontsevich.

关于有界自动群中ET0L子群成员问题

On the ET0L subgroup membership problem in bounded automata groups

本文研究了有界自动群中无限射线稳定子群的成员问题，证明其形成可构造的ET0L语言，并给出了生成函数的递归公式。

Boundary action of automaton groups without singular points and Wang tilings

We study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk. We show that, in general, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Bas…

Bounding the minimal number of generators of groups and monoids of cellular automata

For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we give upper bounds for the rank in terms of the number of conjugacy classes of subgroups of $G$. The case when $G$ is a finite cyclic group has been studied before, so here we focus on the cases when $G$ is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lowe…

Cellular Automata and Finite Groups

For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of groups that depends on the numbers $α_{[H]}$ of periodic configurations for conjugacy classes $[H]$ of subgroups of $G$. …

An analogue to Dixon's theorem for automaton groups

Dixon's famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that it is hopeless to wish for a uniform distribution -- or even a non-trivial one -- by drawing random permutations and looking at the generated group. Mealy automata are a powerful tool to generate groups, including all finite groups and many interesting infinite ones, whence the idea of generating random finite groups by drawing random Mealy automata. In thi…