Related papers: A sharp diameter bound for unipotent groups of cla…

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…

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…

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…

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)…

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 $\Gamma (G,S)$ will have a diameter of $(\log |G|)^{O(1)}$. However, little progress has…

Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set…

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.

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.

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…

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…

Let $S\subset\text{GL}_n(\mathbb Z)$ be a finite symmetric set. We show that if the Zariski closure of $\Gamma=\langle S\rangle$ is a product of $\text{SL}_d$ or a special affine linear group, then the diameter of the Cayley graph…

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 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…

Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any…

Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of…

Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the…

We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szab\'o (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting…

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…

The directed Cayley diameter of a finite group is investigated in terms of the monoid of product-one sequences over the group, via the new notion of directed geodesic atoms. Two quantities associated to the set of directed geodesic atoms…

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup…