Sean Eberhard, Urban Jezernik
Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability $1 - o(1)$. In the particular case $G = \mathrm{SL}_n(p)$ with $p$ a prime of bounded size, we show that the same holds for $k = 3$.
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The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive coherent configuration with $n$ vertices is not a Cameron scheme, then its automorphism group has minimal degree $\geq cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this pap…
Given a non-abelian finite simple group $G$ of Lie type, and an arbitrary generating set $S$, it is conjectured by Laszlo Babai that its Cayley graph $Γ(G,S)$ will have a diameter of $(\log |G|)^{O(1)}$. However, little progress has been made when the rank of $G$ is large. In this article, we shall show that if $G$ has rank $n$, and its base field has order $q$, then the diameter of $Γ(G,S)$ would be $q^{O(n(\log n + \log q)^3)}$.
Diameter of classical groups generated by transvections
本文证明由平延生成的有限经典群在含平延生成集下 Cayley 图直径为 $(n\log q)^C$，并结合随机生成元结果证实相关 Babai 猜想。
Random generators of the symmetric group: diameter, mixing time and spectral gap
令$g$、$h$是$G=Sym(n)$或$G=Alt(n)$的随机生成元对。我们表明，随着概率趋于$1$作为$n\to \infty$，（a）$G$相对于$S = \{g,h,g^{-1},h^{-1}\}$的直径至多为$O(n^2 (\log n)^c)$，（b）$G$相对于$S$的混合时间至多为$O(n^3 (\log n)^c)$。(Both $c$和隐含的常数是绝对的。）这些界限远低于已知的最强最坏情况界限（Helfgott-Seress，2013）;它们大致匹配已知的最坏例子。我们还给出了一个改进的，但仍然是非常数，约束的光谱间隙。我们的结果依赖于（Babai-Beals-Seress，2004）中的算法和以下事实的组合：一对随机置换的作用几乎肯定充当$\ell$ -元组上的扩展器，其中$\ell$是任意常数（Friedman等人，1998年）。
For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n {(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm…
$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(Γ)$ of a lattice $Γ$ in a high rank simple Lie group $H$ grows sub-linearly with $v = μ(H / Γ)$, the co-volume of $Γ$ in $H$. We prove this for non-uniform lattices in a very strong form, showing that for $2-$generic such $H$'s, $d(Γ) = O_H(\log v / \log \log v)$, which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than $d(Γ) = O(\log v)$.
The Gilman-Maskit algorithm for determining the discreteness or non-discreteness of a two-generator subgroup of $\mathrm{PSL}_2\mathbb{R}$ terminates with a pair of generators that are Farey words. The Farey words are primitive words that are indexed by rational numbers and infinity. The E-words, primitive words with palindromic product forms, are also indexed by rational numbers and infinity. We show a modification of the Gilman-Maskit algorithm so that the stopping generators are E-words.
For an arbitrary countable group G = <A|R> given by its generators A and defining relations R we discuss a specific method for embedding of G into a certain 2-generator group T. Our embedding explicitly lists the images of generators from A in the group T, and from the relations R it explicitly deduces defining relations for T inheriting certain special features from R. The obtained method can be used in constructions of explicit embeddings of recursive groups into finitely presented groups.
The minimal number of generators of a finite semigroup
有限半群的秩是生成该半群所需的最小元素数。给出了群上任意（不一定正则）Rees矩阵半群的秩的一个公式。该公式用结构矩阵的维数和由结构矩阵中的元素生成的子群得到的结构群的某个子集的相对秩来表示，并假设该结构矩阵是Graham规范型的.然后应用这个公式回答了某些自然变换半群族的极小生成集问题。特别是，确定的问题的最大秩的子半群的全变换么半群（和对称逆半群）被认为是。
Embeddings using universal words in the free group of rank 2
本文提出一种将可数群G=<A|R>嵌入2-生成元群T的简单方法，使生成元像明确且T的定义关系可自动推导，并展示了具体应用与递归群嵌入的参考。
We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups
A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations
本文通过生成元与关系对秩二复反射群进行组合推广，提出 J-reflection groups 的一致展示，证明其中心为循环群，并完成反射同构分类。
Generators for Arithmetic Groups
证明了高阶半单李群中的非余紧不可约格都含有一个有限指标子群，它有三个生成元。
Actions of the n-strand braid groups on the free group of rank n which are similar to the Artin representation
我们对$n$ -股辫群在秩为$n$的自由群上的作用进行了分类，这与Artin表示类似，因为$B_{n}$的第$i$个生成元$σ_{i}$的作用使得它固定了除$j = i,i+1$之外的所有自由生成元$x_{j}$。我们确定所有这样的表示和讨论结不变量来自这样的表示。
On the rank of a Coxeter group
设W是具有Coxeter生成元S的Coxeter群。Coxeter系统（W，S）的秩是S的基数|S|。Coxeter系统（W，S）有有限秩当且仅当W是有限生成的.如果（W，S）具有无限秩，则|S| = |W|，因为W的每个元素都由S的元素的有限积表示。如果W不是单生的，则（W，S）的秩由W唯一确定。如果W是随机生成的，则W可以具有不同秩的Coxeter生成器S和S'的集合。本文确定了任意n-生成Coxeter群W的所有可能秩的集合。
A finite permutation group $G$ on $Ω$ is called a rank 3 group if it has precisely three orbits in its induced action on $Ω\times Ω$. The largest permutation group on $Ω$ having the same orbits as $G$ on $Ω\times Ω$ is called the 2-closure of $G$. We construct a polynomial-time algorithm which given generators of a rank 3 group computes generators of its 2-closure.
Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds
本文提出了一种统一的显式构造方法，为所有秩至少为三的整数域上的特殊线性群提供有限二生成表示，包含多项式复杂度界及多种扩展应用。
Universal lattices and unbounded rank expanders
我们研究了非交换泛格的表示，并利用它们来计算交换泛格$G_{d,k}= \SL_d(\Z[x_1,...,x_k])$关于几个生成集的\TauC的下界。作为上述结果的一个应用，我们证明了有限群$\SL_{3k}(\F_p)$的Cayley图可以通过适当地选择生成元而成为扩张图.这提供了第一个例子的扩展家庭的团体的李型的秩是没有界的，并给出了一个自然的（和明确的）反例，两个programmures的亚历克斯Lubotzky和本杰明韦斯。
The congruence subgroup problem for a finitely generated group $Γ$ asks whether $\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $Γ=F_{2}$ and $Φ_{2}$) and a new result for $Γ=Φ_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(Φ_{2}\right)=\hat{F}_ω$ when $Φ_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_ω$ is the free pr…
We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product g=g_1 g_2 ... g_k such that the cyclic group generated by each g_i is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many ge…
We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $(\mathbb Z_2^2)\wr\mathbb Z$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring $\mathbb Z_2[[t]]$ of formal power series over $\mathbb Z_2$.
Homogeneous number of free generators
我们回答西蒙·托马斯的两个问题。首先，我们证明了对于任意n>2，可以找到SLn（Z）的一个四生成自由子群，它是概率稠密的。更一般地，我们证明了一个允许同余子群性质的算术群Γ，有一个其秩有一个显式界的伪稠密自由子群。其次，我们证明了这样一个算术群Γ的局部自由子群的集合是不可数的。
The rank gradient from a combinatorial viewpoint
本文研究了剩余有限群中有限指数子群的生成元的最小数目的渐近性态。我们分析了三类自然群：顺从群，具有无限可解正规子群的群和虚拟自由群。作为一种工具，我们推广了Lackenby的可分性定理的可分性的情况下，提出了群。
Diameter bounds for arbitrary finite groups and applications
研究提出一个基于群的composition factors直径和最大指数的强界，应用于可解群、anabelian群、transitive soluble subgroups及Grigorchuk猜想等，证明了多项式和对数直径的界。
Principal congruence subgroups in the infinite rank case
本研究将1960年代关于整数一般线性群与特殊线性群的经典结果推广到无限生成自由阿贝尔群的自同构群，给出其正规生成元、极大正规子群分类、主同余子群正规生成元及Brenner梯关系的类比结论。
Growth in finite simple groups of Lie type of bounded rank
本文证明了：如果L是有限的李型单群，A是L的生成元的对称集，则A是增长的，即|AAA| > |A| ^{1+ n}，其中n只与L的李秩有关，或AAA=L.这意味着对于一类秩有界的李型单群L，任何Cayley图的直径在|L|中是多对数的.我们得到了GL（n，p）的p阶完全子群的Cayley图的直径的一个类似的界，并得到了一些新的扩张子族.我们还证明了下面的部分扩展。设G是GL（n，p）的子群，p是素数，S是G的对称生成元集，满足|S^3| \le K |S|对于某些K.则G有两个正规子群H \ge P使得H/P是可解的，P包含在S ^6中且S被H的K^c陪集覆盖其中c依赖于n。对于GL（n，F），F是任意域的无穷子群的生成集，我们得到了类似的结果.
JSJ decompositions and polytopes for two-generator one-relator groups
本文建立秩为2阿贝尔化的双曲双生成元单关系群的$Z_{max}$-JSJ分解与Friedl–Tillmann多面体的直接联系，并导出二次时间算法等结构性质。
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…
For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{End}(A^G)$ and $\text{Aut}(A^G)$. In the first part, when $G$ is a finite group, we give upper and lower bounds for the rank of $\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of $G$. In the second part, we apply our bounds to show that if $G$ has an infinite descending chain of normal subgroups of finite index, t…
On an algorithm to decide whether a free group is a free factor of another
我们重新讨论了一个给定的自由群F的一个自由生成子群H是否是F的一个自由因子的问题。已知的算法在时间多项式中解决这个问题，在H的生成元的长度和在F的秩中的指数的长度之和中。我们表明，后者的依赖性可以指数的秩差秩（F）-秩（H），这往往会使一个显着的变化。
Matching theorems for systems of a finitely generated Coxeter group
在本文中，我们证明了一系列的匹配定理的两个集合的Coxeter生成器的一个非线性生成的Coxeter群，确定了两个集合的生成器的共同特征。作为一个应用，我们描述了一个算法，找到一组的Coxeter生成器的最大秩的一个Coxeter群。
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happen…
Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of…
A $p$-group with positive Rank Gradient
我们为$d\geq 2$和$ε>0$构造了一个由$d$生成的$p$ -群$Γ$，它在渐近意义上的行为几乎类似于由$d$生成的自由亲$p$ -群.我们证明了指数$p^n$的子群需要$(d-ε)p^n$生成元，并且$Γ$的子群增长满足$s_{p^n}(Γ)>s_{p^n}(F_d^p)^{1-ε}$，其中$F_d^p$是$d$生成的自由亲$p$ -群.为了做到这一点，我们引入了一个新的不变量，并研究了它的一些基本性质。
A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prüfer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.
Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$ had ranked at least two for all $p, q > 5$ and two independent points. This paper shows that two points that can be extended to a basis for $E_{(p,q)}$ under conditions are confident that we will fully recover.
Characteristically simple Beauville groups, II: low rank and sporadic groups
Beauville曲面是一般类型的刚性复曲面，通过称为Beauville群的有限群的自由作用而同构于更高的乘积。这里我们考虑哪些特征性的单群可以是Beauville群。本文证明了：如果G是单群L_2（q），L_3（q），U_3（q），Sz（2^e），R（3^e）或散在单群的一个幂，则G是Beauville群当且仅当它有两个生成元且不同构于A_5.
The 4-string Braid group $B_4$ has property RD and exponential mesoscopic rank
证明了4弦上的辫群$B_4$及其中心商$B_4/< z>$具有Haagerup-Jolissaint性质RD.因此，2个生成元上的自由群$F_2$的自同构群$\Aut(F_2)$具有性质RD。我们还证明了辫群$B_4$是一个中等秩群（维数为3）。也就是说，我们证明了$B_4$和它的中心商都具有指数介观秩，即，它们包含了指数级的许多不包含在平面中的大平面球。
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