Finite Groups of Uniform Logarithmic Diameter

Logarithmic bounds for the diameters of some Cayley graphs

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 $S$.

Uniform Diameter Bounds in Branch Groups

Let $G$ be either the Grigorchuk $2$-group or one of the Gupta-Sidki $p$-groups. We give new upper bounds for the diameters of the quotients of $G$ by its level stabilisers, as well as other natural sequences of finite-index normal subgroups. Our bounds are independent of the generating set, and are polylogarithmic functions of the group order, with explicit degree. Our proofs utilize a version of the profinite Solovay-Kitaev procedure, the branch structure of $G$, and in certain cases, results on the lower central series of $G$.

New Uniform Diameter Bounds in Pro-$p$ Groups

We give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-$p$ domain (with exponent depending on the dimension); Chevalley groups over a pro-$p$ domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some cons…

Uniform poly-log diameter bounds for some families of finite groups

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 $n$ and any generating set $S\subseteq G_{n}$, \[ diam(G_n,S) \leq C \cdot log^d (|G_n|)\] when $diam (G,S)$ is the diameter of the finite group $G$ with respect to the set of generators $S$. It is defined as the maximum over $g \in G$ of the length of the shortest word in $S \cup S^{-1}$ representing $g$. This result shows that these families of finite groups…

具有一致几乎平坦商群的剩余有限群

Residually finite groups with uniformly almost flat quotients

本文证明多循环群及有限生成剩余无挠幂零群，若其有限商群直径一致地以大小为多项式的下界，则群为虚拟幂零，推广了Khukhro和Valette的结果。

Short Laws for Finite Groups of Lie Type

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki groups. We also produce short laws valid for generating pairs and random pairs in finite groups of Lie type, and, conditional on Babai's diameter conjecture, make effective the dependence of our bounds on the rank. Our proof uses, among other tools, the Classification of Finite Simple Groups, Aschbacher's structure theorem for maximal subgroups for classical group…