arXiv subjects

Henry Bradford

Publications and source records attributed to Henry Bradford.

23 records · Page 2Linked to original sources

arXiv2017

Constructing Directed Cayley Graphs of Small Diameter: A Potent Solovay-Kitaev Procedure

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 words, but also only constructs small-diameter undirected Cayley graphs. As an application we give directed diameter bounds on finite quotients of two very different groups: $SL_2 (\mathbb{F}_q [[t]])$ (for $q$ even) and a group of automorphisms of the ternary rooted tree introduced by Fabrykowski and Gupta.

math.GR↗

arXiv2017

Short Laws for Finite Groups and Residual Finiteness Growth

We prove that for every $n \in \mathbb{N}$ and $δ>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+δ}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom, About the length of laws for finite groups, Isr. J. Math.]. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.

math.GR↗

arXiv2017

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 $G$.

math.GR↗

arXiv2015

Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p [t])$

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of $SL_2 (\mathbb{F}_p [t])$ modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape of such walks from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in $SL_2 (\mathbb{F}_p [t])$.

math.GR↗

arXiv2014

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 consequences of our results for random walks on groups.

math.GR↗