Geometric group theory and arithmetic diameter

TL;DR

This paper explores the properties of metric spaces derived from groups with infinite diameter, focusing on the structure and growth of elements with fixed word length, especially in groups generated by integers with prime factor restrictions.

Contribution

It proves that certain groups generated by integers with restricted prime factors have infinite diameter and discusses the open problem of computing the minimal element for given word lengths.

Findings

Infinite diameter for groups generated by integers with restricted prime factors.

Existence of infinitely many elements at each fixed distance from the identity.

Open problem on computing the minimal element for given word lengths.

Abstract

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let \lambda_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute \lambda_A(h) and estimate its growth rate.