A well-known conjecture of Babai states that if $G$ is any finite simple group and $X$ is a generating set for $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded by $\log|G|^c$ for some universal constant $c$. In this paper, we prove such a bound for $Cay(G,X)$ for $G=PSL(n,q),PSp(n,q)$ or $PSU(n,q)$ where $q$ is odd, under the assumptions that $X$ contains a transvection and $q\neq 9$ or $81$.
A well-known conjecture of Babai states that if $G$ is a finite simple group and $X$ is a generating set of $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded above by $(\log |G|)^c$ for some absolute constant $c$. The goal of this paper is to prove such a bound for the diameter of $Cay(G,X)$ whenever $G=SL(n,p)$ and $X$ is a generating set of $G$ which contains a transvection. A natural analogue of this result is also proved for $G=SL(n,K)$, where $K$ can be any field.
Based on the rules of magic cubes, a game of two-dimensional magic cube was deliberately designed. This essay will explore its properties with the assistance of group theory and computer programming. It will first elaborate the rules of two-dimensional magic cube and then use group theory to comprehensively study its properties like the order of the permutation group of the cube, the diameter of Cayley Graph, etc. At the last part of the paper, a much more general result will be raised to satisfy situations like the irregular magic cubes.
We show that for integers k > 1 and n > 2, the diameter of the Cayley graph of SL_n(Z/kZ) associated to a standard two-element generating set, is at most a constant times n^2 ln k. This answers a question of A. Lubotzky concerning SL_n(F_p) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL_n(Z/kZ).
Let $Γ$ be a group and $(Γ_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $Γ/Γ_n$ , under some natural hypotheses on the behaviour of power and commutator words in $Γ$. The bounds we obtain do not depend on a choice of generating set. Moreover under reasonable conditions our method provides a fast algorithm for constructing directed Cayley graphs of diameter satisfying our bounds. The proof is closely analogous to the the Solovay-Kitaev procedure, which only uses commutator …
Let $S\subset\text{GL}_n(\mathbb Z)$ be a finite symmetric set. We show that if the Zariski closure of $Γ=\langle S\rangle$ is a product of $\text{SL}_d$ or a special affine linear group, then the diameter of the Cayley graph $\text{Cay}(Γ/Γ(q),π_q(S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, $π_q:Γ\to Γ/Γ(q)$ is the canonical projection induced by the reduction modulo $q$, and the implied constant depends only on $S$.
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, …
In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.
We give an example of an infinite family of finite groups $G_n$ such that each $G_n$ can be generated by 2 elements and the diameter of every Cayley graph of $G_n$ is $O(\log (| G_{n}|))$. This answers a question of Lubotzky.
The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n^2 log p, where n is the rank of the associated Lie group. This generalizes a result of the first author, which treated the case of SL_n(Z/pZ). (Keywords: diameter, Cayley graph, finite groups of Lie type. AMS classification: 20G40, 05C25)
We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.
We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.
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 $q^{O(n(\log n + \log q)^3)}$.
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is generally easy to find counterexamples to such regularity results when the assumption of local finiteness is dropped. The present work focuses on the following problem: determining whether these regularity properties still hold when local finiteness is replaced by an assumption of transitivity. After recalling the locally finite situation, we show that there…
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)
Institut Curie, CNRS UMR168, Paris, France（居里研究所，法国国家科学研究中心UMR168，法国巴黎）；University of Washington（华盛顿大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡数学研究中心，三峡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Independent Researcher（独立研究员）；Moscow State University（莫斯科国立大学）；Kazakh-British Technical University（哈萨克-英国技术大学）；Neapolis University Pafos, Cyprus（尼阿波利斯大学帕福斯校区，塞浦路斯）；City St George's, University of London（伦敦大学城市圣乔治学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；ITMO University（ITMO大学）；Moscow Center for Advanced Studies（莫斯科高等研究中心）；National Research University Higher School of Economics（国立研究大学高等经济学院）；Ariel University（阿里埃勒大学）；Saint-Petersburg Pasteur Institute（圣彼得堡巴斯德研究所）；Criteo AI Lab, Berlin, Germany（Criteo人工智能实验室，德国柏林）；University of Bologna（博洛尼亚大学）；Faculty of Bioengineering and Bioinformatics, Lomonosov Moscow State University（莫斯科国立大学生物工程与生物信息学系）；Researcher at Banner Stat（Banner Stat研究员）；MIREA — Russian Technological University（MIREA — 俄罗斯技术大学）；Research Center of the Artificial Intelligence Institute, Innopolis University（因诺波利斯大学人工智能研究所研究中心）；Inpdependent Researcher（独立研究员）；LIX, Ecole Polytechnique/CNRS, IP Paris（LIX，巴黎综合理工学院/法国国家科学研究中心，巴黎理工学院）
We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p…
By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a pro…
We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with $G=\mbox{SL}_{n}({\mathbb F}_{q})$ ($q>2$) acting on $k$-tuples of vectors in ${\mathbb F}_{q}^{n}$; moreover, an exact expression is found in the case of Cayley graphs.
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the identity in a finitely presented group can either be measured intrinsically its 1-skeleton or extrinsically in the Cayley graph of the group. We construct the first examples of closed manifolds and finitely presented groups for which this choice -- intrinsic versus extrinsic -- gi…
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order $n^2$ to the sphere of radius $n$. We also discuss the rarity of d…
We study the relation between the diameter, the first positive eigenvalue of the discrete $p$-Laplacian and the $\ell_p$-distortion of a finite graph. We prove an inequality relating these three quantities and apply it to families of Cayley and Schreier graphs. We also show that the $\ell_p$-distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain estimates for the convergence to zero of the spectral gap as an application of the main result.
We give a non-deterministic algorithm that expresses elements of SL_N(Z), for N > 2, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the non-deterministic time-complexity of the subtractive version of Euclid's algorithm for finding the greatest common divisor of N > 2 integers a_1,..., a_N is at most a constant times N log n where n := max {|a_1|,..., |a_N|}. This leads to an elementary proof that for N > 2 the word metric in SL_N(Z) is biLipschitz equivalent to the logarithm of the matrix norm -- an instance of a th…
We provide an explicit construction of finite 4-regular graphs $(Γ_k)_{k\in \mathbb N}$ with ${girth Γ_k\to\infty}$ as $k\to\infty$ and $\frac{diam Γ_k}{girth Γ_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For each fixed dimension $n\geqslant 2,$ we find a pair of matrices in $SL_{n}(\mathbb{Z})$ such that (i) they generate a free subgroup, (ii)~their reductions $\bmod\, p$ generate $SL_{n}(\mathbb{F}_{p})$ for all sufficiently large primes $p$, (iii) the corresponding Cayley graphs of $SL_{n}(\mathbb{F}_{p})$ have girth at least $c_n\log p$ for some $c_n>0$. Relying on growth resul…
We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph G of large diameter admits a quotient with fibres of small diameter on which the action of Aut(G) is virtually abelian with vertex stabilisers of bounded size. We also show that G has moderate growth in the sense of Diaconis and Saloff-Coste, …
We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitti…
We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain,Gamburd and Varjú it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for square-free moduli m which are relatively prime to some number m_0 the Cayley graphs Ga…
Let G be a Chevalley group scheme of rank l. We show that the following holds for some absolute constant d>0 and two functions p_0=p_0(l) and C=C(l,p). Let p>p_0 be a prime number and let G_n:=G(\Z/p^n\Z) be the family of finite groups for n>0. Then for any n>0 and any subset S which generates G_n we have diam(G_n,S)< C n^d, i.e., any element of G_n is a product of Cn^d elements from S\cup S^{-1}. In particular, for some C'=C'(l,p) and for any n>0 we have, diam(G_n,S)< C' log^d(|G_n|). Our proof is elementary and effective, in the sense that the constant d and the functions p_0(l) and C(l,p) a…
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…
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(Γ(G,A))$ of the Cayley graph $Γ(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam(G):= $\max_A$ diam$(Γ(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$, but the best previously known upper bound was exponential in $\sqrt{n \log n}$. We give a quasipolynomial upper bound, namely, \[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \e…
The diameter of a finite group $G$ with respect to a generating set $A$ is the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A \cup A^{-1}$. We denote this invariant by $\diam_A(G)$. It can be interpreted as the diameter of the Cayley graph induced by $A$ on $G$ and arises, for instance, in the context of efficient communication networks. In this paper we study the diameters of a finite abelian group $G$ with respect to its various generating sets $A$. We determine the maximum possible value of $\diam_A(G)$ and classify…
We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.
Benjamini, Finucane and the first author have shown that if (G_n,S_n) is a sequence of Cayley graphs such that |S_n^n|=O(n^D|S_n|), then the sequence (G_n,d_{S_n}/n) is relatively compact for the Gromov-Hausdorff topology and every cluster point is a connected nilpotent Lie group equipped with a left-invariant sub-Finsler metric. In this paper we show that the dimension of such a cluster point is bounded by D, and that, under the stronger bound |S_n^n|=O(n^D), the homogeneous dimension of a cluster point is bounded by D. Our approach is roughly to use a well-known structure theorem for approxi…
We define the positive diameter of a finite group $G$ with respect to a generating set $A\subset G$ to be the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A$. This invariant, which we denote by $\diam_A^+(G)$, can be interpreted as the diameter of the Cayley digraph induced by $A$ on $G$. In this paper we study the positive diameters of a finite abelian group $G$ with respect to its various generating sets $A$. More specifically, we determine the maximum possible value of $\diam_A^+(G)$ and classify all generating sets…
We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon is an absolute positive real number, or AAA=G. As a corollary we get that the diameter of any Cayley graph of G is Poly-Logarithmic in |G|.
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 $k = 3$.
Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,...,n\}$ and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$ the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph $A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this paper, we intensively investigate the properties of $A\T_n^q$ including connectedness, diameter, independence number, clique number, chromatic number and the maximum-size independent sets of $A\T_n^q.$ By using the result on the maximum-size independent sets $A\T_n^q$, we completely determine the …
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale version of weak diameter colorings (also known as weak diameter network decompositions), i.e. colorings in which each monochromatic component has small weak diameter. In this paper, we prove that for any $p$, the class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2. This implies that the class of all graphs embeddable on any fixed surf…
Generating the symmetric group by three prefix reversals
Indiana University（印第安纳大学）；Krasovskii Institute of Mathematics and Mechanics（克拉索夫斯基数学与力学研究所）；Ural Mathematical Center（乌拉尔数学中心）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡大学三峡数学研究中心）
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$ (ie $1 \ll k = |G|^{o(1)}$). A conjecture of Aldous and Diaconis (1985) asserts, for $k\gg\log|G|$, that the random walk on this graph exhibits cutoff. When $\log k \lesssim \log\log|G|$ (ie $k = (\log |G|)^{\mathcal O(1)}$), the only example of a non-Abelian group for which cutoff has been established is the dihedral group. We establish cutoff (as $p\to infty$) for the group of $d \times d$ unit upper triangular matrices with integer entries modulo…
